Basic Group Theory II

1
Basic Group Theory II
2
6. Subgroups
3
6. Subgroups

• In the Marching Group we can limit our commands
  to two "Attention!" and "About Face!" That would
  give us a group with the Cayley table as follows

4
6. Subgroups

• But we have seen this little table before. It was
  just the northwest corner of one of our Cayley
  tables for the whole Marching Group

5
6. Subgroups

• This is an example of a subgroup
• a group within a group.
• A subset S of a group G is a subgroup of G if
• S is closed under the group operation of G,
• if the identity element of G is in S and
• for every a in S, a-1 is also in S.

6
6. Subgroups

• The definition specifically requires that three
  of the four group axioms be satisfied for the
  subgroup. The axiom that gets in free is the
  associative property. Since we are using the
  group operation of the "larger" group the
  associative property for all elements of the
  larger group is assumed . Thus the restriction to
  the elements of the subgroup cannot introduce a
  violation of the associative axiom.

7
6. Subgroups

• Example
• Consider the group of integers under the
  operation of addition. If we look at the subset
  of integers that are divisible by 17 we can show
  that this subset is a subgroup of the group of
  integers. If we add two numbers divisible by 17
  together we ge a number that is divisible by 17.
  Closure is satisfied. 0, the additive identity
  for the integers is divisible by 17 so the
  identity property is satisfied. Finally if M is
  divisible by 17 then so is -M and we find that
  the inverse property is satisfied. With all three
  conditions satisfied the set of integers
  divisible by 17 forms a subgroup of the additive
  group of integers.

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6. Subgroups

• There was nothing special about the number 17 in
  the above example. We could repeat the same
  reasoning with any other integer such as 2 or 34
  or 31415926535. Thus we know that the set of all
  mulitples of a given n is a subgroup of the group
  of integers under addition.
• Each group of two or more members has at least
  two subgroups. The subgroup containing the
  identity element only and the group itself. These
  are called trivial subgroups. Any subgroups of a
  group G other than the two trivial subgroups are
  nontrivial subgroups.

9
6. Subgroups

• THEOREM A finite subset of a group that is
  closed under the group operation is a subgroup of
  that group
• This theorem states that for a finite subset of a
  group we need only check for closure to know that
  it is a subgroup. To prove this theorem we,
  therefore, need to show that for a finite subset
  of a group closure implies that the identity
  axiom and inverse axiom are also satisfied.

10
6. Subgroups

• THEOREM If S is a subset of a group G and if for
  every a and b in S the group product a b-1 is
  in S then S is a subgroup of G.
• By the first theorem a finite group requires only
  closure under the group operation to be a
  subgroup. By the second theorem any subset is a
  subgroup if it is closed under taking an inverse
  then multiplying by a set element.

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6. Subgroups

• THEOREM If U is a collection of subgroups of a
  group G then the intersection of U is also a
  subgroup of G.

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7. Cosets
13
7. Cosets

• Consider the subgroup of integers divisible by 3.
  This forms a subgroup of the additive group of
  integers. Its elements are . . . -9, -6, -3, 0,
  3, 6, 9, . . .. By adding 1 to any multiple of 3
  we get another subset of group of integers, . .
  . -8, -5, -2, 1, 4, 7, 10, . . .. By adding 2 to
  the elements of the multiples-of-3 subgroup gives
  us the subset . . . -7, -4, -1, 2, 5, 8, 11, .
  . .. These three subsets exhaust the integers.
  Any integer will be in one and only one of these
  subsets.

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7. Cosets

• One interesting point about the three subsets. If
  you take any two elements, a and b out of one of
  the subsets (both from the same subset) then
  their difference, a - b, will be a multiple of
  three. Their difference will be in the subgroup
  that originally generated them.

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7. Cosets
Another example occurs in the Marching Group.

• A and B form a subgroup as we have already seen.
  R and L form a coset of that subgroup. The
  property mentioned in the multiple-of-three
  example holds here. If any member of R, L is
  multiplied by the inverse of any member of R, L
  the result is in the A, B subgroup.

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7. Cosets

• Let's look at another example, the group of
  symmetry movements of the equilateral triangle.
  Its elements are, in the notation we saw earlier,
  e, a, b, X, Y, Z.

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7. Cosets

• A glance at the north-west corner of the Caley
  table for this group shows that the subset e, a,
  b is closed under the group operation followed
  by. Since it is a finite subset this suffices for
  it to be a subgroup. A coset of the subgroup is
  X, Y, Z. Note again that any member of this
  coset multiplied by the inverse of any member
  gives an element of the subgroup.

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7. Cosets

• If you look at the table again you will notice
  that the subset e, X is also closed under the
  group operation. As it is finite we know that it
  also is a subgroup of the main group. Cosets of
  this group are Y, b and Z, a. These cosets
  obey the property that if a and b are in the
  coset then a b-1 is in the subgroup. But what
  about a-1 b? Let's try the coset Y, b Note
  that
• b-1 Y a Y Z
• which is not in the subgroup. However the cosets
  Y, a) and Z, b obey the property that if u and
  v are in the coset then u-1 v is in the
  subgroup. Thus the cosets with the u v-1
  -is-in-the-subgroup property and the cosets with
  the u-1 v-is-in-the-subgroup property are
  different cosets.

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7. Cosets

• The cosets that have the u v-1
  -is-in-the-subgroup property are called right
  cosets and the cosets that have the u-1
  v-is-in-the-subgroup property are called left
  cosets.
• Subgroups whose right cosets are also left cosets
  are very important in group theory. They are
  called normal subgroups.

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7. Cosets

• Definition
• If H is a subgoup of a group G then for any
  element g of the group the set of products of the
  form h g where h is in H is a right coset of H
  denoted by the symbol Hg. The set of all products
  of the form g h where h is in H is a left coset
  of the subgroup H denoted by the symbol gH.
• In fact if we take any subset U of a group G
  (which can be a subgroup or not) we can multiply
  every element in it from the right by some group
  element v to get another subset V of G. If the
  subset H happens to be a subgroup of G then the
  subset it is transformed into is called a right
  coset of the subgroup H.

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7. Cosets

• In fact if we take any subset U of a group G
  (which can be a subgroup or not) we can multiply
  every element in it from the right by some group
  element v to get another subset V of G. If the
  subset H happens to be a subgroup of G then the
  subset it is transformed into is called a right
  coset of the subgroup H.

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7. Cosets

• In set-builder notation we can define Uv as
• Uv u v u is in U
• When U is a subgroup this is a right coset of U.
• The operation of "multiplication on the right by
  a group element" which transforms subsets into
  subsets (and subgroups into right cosets) will
  also transforms any right coset of a subgroup H
  into a right coset of H (This could be a
  different right coset than we started with or the
  same one). This follows from the definition since
  (Hu)v is all elements of the form (h u) v
  which by the associative property is the same as
  the set of elements of the form h (u v) which
  is none other than H(u v).
• Of course similar stuff is true of left cosets.
  Just hold your brain up to a mirror while doing
  proofs of right-coset properties.

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7. Cosets

• THEOREM If x and y are in the same right coset
  of a subgroup H then x y-1 is in H.
• THEOREM If x y-1 is in subgroup H of group G
  then x and y are in the same right coset of H.

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8. Lagrange's Theorem
25
8. Lagrange's Theorem

• Definition
• If G is a finite group (or subgroup) then the
  order of G is the number of elements of G.

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8. Lagrange's Theorem

• Lagranges Theorem
• The order of a subgroup H of group G divides the
  order of G.

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8. Lagrange's Theorem

• Lagranges Theorem
• One of the immediate results of Lagrange's
  Theorem is that a group with a prime number of
  members has no nontrivial subgroups.

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8. Lagrange's Theorem

• Lagranges Theorem
• A consequence of Lagrange's Theorem would be, for
  example, that a group with 45 elements couldn't
  have a subgroup of 8 elements since 8 does not
  divide 45. It could have subgroups with 3, 5, 9,
  or 15 elements since these numbers are all
  divisors of 45.

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8. Lagrange's Theorem

• Lagranges Theorem
• Lagrange's Theorem simply states that the number
  of elements in any subgroup of a finite group
  must divide evenly into the number of elements in
  the group. Note that the A, B subgroup of the
  Atayun-HOOT! group has 2 elements while the
  Atayun-HOOT! group has 4 members. Also we can
  recall that the subgroups of S3, the permutation
  group on 3 objects, that we found cosets of in
  the previous chapter had either 2 or 3 elements
  -- 2 and 3 divide evenly into 6.

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8. Lagrange's Theorem

• Lemma If H is a finite subgroup of a group G and
  H contains n elements then any right coset of H
  contains n elements.
• Lemma Two right cosets of a subgroup H of a
  group G are either identical or disjoint.
• Lemma The number of elements in each equivalence
  class is the same as the number of elements in H.

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8. Lagrange's Theorem

• Definition
• If H is a subgroup of G then the number of left
  cosets of H is called the index of H in G and is
  symbolized by (GH). From our development of
  Lagrange's theorem we know that
• G H (GH)

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8. Lagrange's Theorem

• Theorem If the order of a group G is divisible
  by 2 then G has a subgroup of two elements.
• This is the converse of Lagrange's Theorem. One
  of the most interesting questions in group theory
  deals with considering the converse of Lagrange's
  theorem. That is if a number n divides the order
  of group G does that mean that G must have a
  subgroup of order n? The answer is no in general
  but the special cases where it does work out are
  many and interesting. They are dealt with in
  detail in the Sylow Theorems which we will treat
  later.

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9. Cyclic Groups and Subgroups
34
9. Cyclic Groups and Subgroups

• Let's start with the number 1. We'll allow
  ourselves to add or subtract the number 1 to get
  to new numbers.
• Question what integers will we be able to reach
  by this process?
• Answer all of them.
• To get to 17 simply add 1 16 times. To get to -42
  simply subtract 1 43 times. The fact that the
  integers can be "built" by adding and subtracting
  1 means that the additive group of integers is a
  cyclic group.

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9. Cyclic Groups and Subgroups

• Now let's look at the Marching Group. The entire
  Marching Group can be "built" from repeated
  applications of R
• R (R1) R
• R R (R2) B
• R R R (R3) L
• R R R R (R4) A

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9. Cyclic Groups and Subgroups

• Groups that can be generated in their entirety
  from one member are called cyclic groups. For
  infinite groups we have to clarify what we mean
  by "generated from." For example in the additive
  group of Integers starting with 1 and adding it
  over and over to itself will never get a negative
  number nor the identity zero. Thus for a cyclic
  group we have the definition that all the
  elements may be generated from a single element
  together with its inverse. For finite cyclic
  groups the addition of "together with its
  inverse" is not needed. That statement probably
  should be stated as a theorem.

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9. Cyclic Groups and Subgroups

• Theorem If a finite group G can be generated
  from one of its elements, a together with its
  inverse a-1 then it can be generated from a
  alone.

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9. Cyclic Groups and Subgroups

• Just as we do for multiplication of regular old
  numbers we can express repeated application of
  the group operation by a given element with
  exponential notation. So we'll agree that for an
  element a of a group G with operation
• an means a a a . . . a (n times)
• a-n means a-1 a-1 a-1 . . . a-1 (n times)
  
• a0 means the identity element e

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9. Cyclic Groups and Subgroups

• We find that with this notation the basic law of
  exponents all work as in regular arithmetic
  namely
• am an amn
• The inverse of an is a-n
• (am )n amn
• With this notation we mean that a power of a is
  an where n is any integer positive, negative or
  zero.

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9. Cyclic Groups and Subgroups

• Cyclic Subgroups
• If we pick some element a from a group G then we
  can consider the subset of all elements of G that
  are powers of a. This subset forms a subgroup of
  G and is called the cyclic subgroup generated by
  a. It forms a subgroup since it is
• Closed If you multiply powers of a you end up
  with
• powers of a
• Has the identity a a-1 a0 e
• Has inverses The inverse of any product of a's
  is a
• similar product of a-1
  's.

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9. Cyclic Groups and Subgroups

• But this is the long way of proving subgrouphood.
  Let's use our theorem that says if x and y are in
  the subset implies that x y-1 is in the subset
  then the subset is a group. This is simple here.
  If y is a power of a then so is y-1 and so,
  therefore, is x y-1

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9. Cyclic Groups and Subgroups

• A few facts about cyclic groups
• and cyclic subgroups
• Cyclic groups are Abelian.
• All groups of prime order are cyclic.
• The subgroup of a group G generated by a is the
  intersection of all subgroups of G containing a
• All infinite cyclic groups look like the additive
  group of integers.

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9. Cyclic Groups and Subgroups

• We have a one to one correspondence an
  corresponds to n which is preserved by the group
  operation. You can either perform the operation
  in one group and then find the corresponding
  element in the other or you can first find the
  corresponding elements in the other group and
  then perform the operation on them. The result is
  the same. And it works both ways. This is a
  group isomorphism, a concept we have looked at
  but haven't rigorously defined yet.

44
9. Cyclic Groups and Subgroups

• Definition
• Two groups G and H are isomorphic if there exists
  a one-to-one relation f G gt H between them
  such that for any x and y in G, f(x) f(y) f(x
  y). The one-to-one relation f is called an
  isomorphism. Thus two groups are said to be
  isomorphic if an isomorphism exists between them.

45
9. Cyclic Groups and Subgroups

• Note on the notation Two different symbols for
  the group operations are used in the prior slide.
  represents the group operation in H and
  represents the group operation in G. This was to
  emphasize that we are dealing with two different
  groups each with its own group operation.

46
9. Cyclic Groups and Subgroups

• One of the most common examples of a group
  isomorphism is found in the theory of logarithms.
  Let f G gtH be the logarithm function and let G
  be the group of positive real numbers under
  multiplication and let H be the real numbers
  under addition. It turns out that
• Log(x) Log(y) Log(xy)
• This together with the fact that Log is a
  one-to-one function (it has an inverse -- the
  exponential or "antilog" function) makes this a
  group isomorphism. This used to have great
  practical importance back in the days before
  pocket calculators. Long numerical calculations
  involving multiplication (and taking of roots and
  powers) could be simplified by doing addition of
  the Logs of the numbers. In those days no
  scientist or engineer was far from his table of
  Logarithms.

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9. Cyclic Groups and Subgroups

• The following are basic facts about
• group isomorphism
• If f G gt H is a group isomorphism then
• If e is the identity in G,
• then f(e) is the identity in H.
• The inverse of f(a) in H is f(a-1).

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9. Cyclic Groups and Subgroups

• The main point about cyclic groups and
  isomorphism is that all finite cyclic groups of
  the same order are isomorphic. Thus there is only
  one cyclic group (up to isomorphism) of order,
  say, 6. If we let a be a generator then the
  elements of the cyclic group of order 6 are
• a, a2, a3, a4, a5, and a6 e.

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9. Cyclic Groups and Subgroups

• The canonical example of a cyclic group of order
  n is the additive group of integers mod n Z/nZ.
  The cayley table is

50
9. Cyclic Groups and Subgroups

• The set of integers mod n is not a group under
  multiplication because 0 has no inverse there
  is nothing that multiplies 0 to give 1, the
  identity element for multiplication. If n is a
  prime number and we throw out zero the remaining
  elements of Z/nZ does form a group a cyclic
  group.

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9. Cyclic Groups and Subgroups

• Here's the Cayley table for the non-zero elements
  of Z (mod 7)

52
9. Cyclic Groups and Subgroups

• The cyclic nature of the group isn't apparent
  from a glance at the table. However if we
  consider the powers of 5 then we find that 5
  generates the entire group. By rearranging the
  elements of the Cayley table the cyclic nature is
  readily seen.

The above table was rearranged by listing the 0th
power of 5 first, then the first power of 5 then
the 2nd power of 5 etc
53
9. Cyclic Groups and Subgroups

• We can form a cyclic subgroup of any group by
  grabbing an element a of the group an looking at
  the set of all its powers. This gives the
  subgroup generated by a notated as lt a gt. Now by
  Lagrange's theorem the number of elements in lt a
  gt (the order of the subgroup) must be a divisor
  of the order of the group. Also if k is the order
  of lt a gt then k is the least positive integer
  with ak e. For any element of a group the least
  positive integer that gives the identity when the
  element is raised to that power is called the
  order of the element. Thus the order of a cyclic
  subgroup and the order of its generator are the
  same number.

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9. Cyclic Groups and Subgroups

• We might remark at this point that any subgroup
  of a cyclic group must itself be cyclic. If G is
  generated by a and S is a subgroup of G Then S
  must be cyclic. To see this consider all the
  elements of S. They can all be expressed as
  powers of a. Let k be the least positive power of
  a that is in S. If every element is not a power
  of ak then there is some element am where m is
  positive and not a multiple of k. Let p be the
  greatest multiple of k that is less than m. Then
  m - p is less than k. But this means that am
  a-p am-p is in S. This contradicts the choice
  of k as the smallest positive power of a in the
  subgroup. Thus all elements of S are generated by
  ak and S is cyclic.

55
9. Cyclic Groups and Subgroups

• Now if for any element its order divides the
  order of the group then it follows that if n is
  the order of the group then an e for any
  element a of any finite group.

56
9. Cyclic Groups and Subgroups

• Now we have a constrained converse of Lagrange's
  theorem. It is true that for cyclic groups if n
  is a divisor of the order of the group then the
  group has a subgroup of order n. To see why this
  must be so recall that a cyclic group is
  generated by the powers of one of its elements.
  Let's see why this must be for the cyclic group
  of order 12.
• C12 e, a, a2, a3, a4, a5, a6, a7, a8, a9,
  a10, a11

57
9. Cyclic Groups and Subgroups

• Since 3 divides 12 (i.e. 3 4 12) then (a3)4
  a12 e. Thus a3 is an element with order 4. (If
  it were less then a power of a less than 12 would
  equal e). Thus a3 generates a subgroup of order
  4. Similarly a4 generates a subgroup of order 3
  and a2 generates a subgroup of order 6 while a6
  generates a subgroup of order 2. There are always
  the trivial subgroups of orders 1 and 12 (The
  subgroup containing only the identity and the
  whole group considered as a subgroup of itself).

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9. Cyclic Groups and Subgroups

• There is nothing special about the number 12. Let
  a cyclic group have order n. If r is a divisor of
  n then there is some number s such that rs n.
  Let a be an element of the group with order n.
  (it must have one if it is cyclic) Then ar is of
  order s and therefore generates a cyclic subgroup
  of order s while as generates a subgroup of order
  r.

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9. Cyclic Groups and Subgroups

• We can go a bit further in looking at subgroups
  of cyclic groups. Not only does a cyclic group
  have a subgroup of order r for every r that
  divides the order of the group but it has exactly
  one subgroup for each distinct divisor of n. To
  see this suppose that S is a subgroup of a cyclic
  group of order n and a is a generator. Suppose S
  has order r where rs n. Then S is generated by
  one of its elements, ak for some k. This means
  that ak has order r which in turn means that k
  times r equals some multiple of n. Now since rs
  n, s must divide k. Therefore ak is in the group
  generated by as which is of order r. However ak
  generates a group of order r. Thus r elements of
  the subgroup generated by ak are in the subgroup
  generated by as which has only r elements. The
  subgroups must be identical.

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10. Permutations
61
10. Permutations

• The "old shell game" is an example of a
  permutation. Three shells, one containing a pea,
  are in a row in front of the sucker -- er, I
  mean, client. The operator then changes the order
  of the shells and challenges the client to tell
  which one contains the pea. The changing of the
  order constitutes a permutation. In the shell
  game there are 6 possible permutations. The
  operator may do nothing. This would be the
  identity permutation where shell 1 stays in its
  position, shell 2 stays in its position and shell
  3 also stays in its position. We can notate this
  permutation as

62
10. Permutations

• The top row of numbers indicates the starting
  position of a shell. The bottom row indicates the
  final position of the shell. Thus to find where
  an object went under a permutation look up its
  number in the top row and its final place will be
  under that number. Thus if the shell-game
  operator switches shells 1 and 2 we can notate
  this permutation by

63
10. Permutations

• Shell 1 went to the place formerly occupied by
  shell 2 (the 2 in the bottom row is below the 1)
  and shell 2 went to the place formerly occupied
  by shell 1 (the 1 in the bottom row is below the
  2). Since there are six different orders that the
  numerals "1, 2" and "3" can be ordered in the
  bottom row the 1 2 3 in the top row is fixed
  there are 6 permutations of 3 objects.

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10. Permutations

• The main reasons for interest in permutations are
  
• They form a group when we use the operation
  "followed by."
• Understanding them helps to solve Rubik's Cube.
• There is a theorem due to Arthur Cayley that says
  that every group is isomorphic to a permutation
  group.
• Permutations are everywhere whether we know it or
  not.

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10. Permutations

• A given set the permutations forms a group with
  the operation followed by. This is because
• permutations are closed under the operation
  followed by,
• since the operation is "followed by" it is
  associative,
• the "leave everything where it is" permutation is
  the identity and
• every permutation can be undone by "running the
  film backwards" thus all permutations have
  inverses.

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10. Permutations

• In a Cayley table for a group each element
  appears exactly once in each row and exactly once
  in each column (Latin square property). This is a
  "pictorial" representation of the cancellation
  property. If I take a group element a and
  multiply it on the right to every element of a
  group I'll get all the elements of the group back
  again. For any element x of G x a will be in G
  (This is just the closure property).

67
10. Permutations

• Furthermore any element of G will be one of the x
  a's since for any b in G the equation x a b
  has a solution. Thus multiplication on the right
  by a permutes the elements of G. What if we were
  to first permute the elements of G by multiplying
  on the right by a followed by multiplying on the
  right by b? That would give us a new permutation
  which would (by the associative property) be the
  same as multiplying on the right by a b.

68
10. Permutations

• Thus each element of G corresponds to a unique
  permutation of the elements of g. And the
  permutation corresponding to the product of two
  elements of G is the composition under followed
  by of the permutations corresponding to the two
  elements singly. This fits the definition of an
  isomorphism. Thus every group is isomorphic to a
  permutation group. This result is known as
  Cayley's Theorem.

69
10. Permutations

• Rubik's Cube and similar puzzles are examples of
  permutation puzzles. The challenge is to form a
  certain permutation from the composition of a
  limited set of permutations. The permutation you
  are challenged to find is the one that permutes
  everything into the solved state. Exactly which
  permutation that is depends on the initial state.

70
10. Permutations

• Even and Odd Permutations
• An old bar bet involves three glasses on the bar.
  They are in a row in front of the sucker -- er, I
  mean client, but the first and third glasses are
  mouth-down on the bar. Let "D" stand for a
  mouth-down glass and "U" stand for a mouth-up
  glass. Then the client sees the following row of
  glasses

71
10. Permutations

• Even and Odd Permutations
• "The object of the game," you tell him, "is to
  get all the glasses mouth-up in exactly three
  moves. A move consists of turning two glasses
  over together, one in each hand." Then you
  demonstrate by turning over the first two glasses
  to reach

72
10. Permutations

• Even and Odd Permutations
• Then you turn over the 1st and 3rd glasses
• Finally you turn over the first two glasses again
  

73
10. Permutations

• Even and Odd Permutations
• All very clean and simple. You now turn over the
  middle glass and bet your "friend" that he can't
  do what he just saw you do. He, having had a few
  drinks, pulls out his wallet and bets you. He
  begins facing
• and no matter what he tries he can't get them all
  mouth-up in exactly three moves.

74
10. Permutations

• Even and Odd Permutations
• The secret of the trick is parity. The number of
  mouth-up glasses is either even or odd. Turning
  over one glass will change the number of mouth-up
  glasses from odd to even or else from even to odd
  since you will be either adding a mouth-up glass
  or subtracting a mouth-up glass. In the game you
  must turn over two glasses simultaneously and
  therefore the parity, either odd or even of the
  number of mouth-up glasses does not change. Now
  the desired final position has 3 (an odd number)
  of mouth up glasses. The trick is that you start
  with 1 (an odd number) of mouth-up glasses but
  you have your "friend" begin with 2 (an even
  number) of mouth up glasses. He can't win.

75
10. Permutations

• Permutations also have a parity which is changed
  by a transposition. A transposition is a
  permutation which exchanges the place of two
  objects whilst leaving all the other objects
  unmoved. Thus
• is a transposition as it swaps elements 2 and 5
  and does not change the positions of 1, 3 and 4.
  An important fact is that any permutation on n
  objects can be done with a series of
  transpositions. It is a further fact that if it
  can be done in an even number of permutations
  then it can't be done in an odd number and vice
  versa.

76
10. Permutations

• Theorem Every permutation is equivalent to a
  product of transpositions.
• Exercise Express the permutation
• as the product of transpositions.

77
10. Permutations

• The second fact is that if a permutation can
  result from an odd number of transpositions then
  it can't result from an even number of
  permutations and vice versa. To see this we will
  define the parity of a permutation For each
  element count up the number of elements that were
  before it in the order before the permutation
  which are after it after the permutation has been
  applied. For instance in the permutation

78
10. Permutations

• there are no elements before 1 before the
  permutation so the total for 1 is zero. There are
  two elements after 4 that were before 4 to begin
  with, namely 2 and 3 so the total for 4 is 2.
  Continuing in this manner we find that the total
  for 6 is 3, the total for 3 is 1, the total for 5
  is 1, the total for 2 is zero, and the total for
  7 is zero. Adding up these totals we get a grand
  total
• If the grand total is an odd number then the
  permutation is of odd parity. If the grand total
  is an even number then the permutation is of even
  parity.
• 7 is an odd number so the example is an odd
  permutation. If you apply any transposition to
  this permutation you will end up with an even
  permutation. (Try it!).

79
10. Permutations

• Why is this so? Why does each transposition
  change the parity of a permutation? Let's look at
  what a transposition does to the parity. Let the
  following represent an ordering of the n numbers
  1 to n.
• Each number in the string of numbers can affect
  the grand total in two ways. First when it is the
  number we're using to count how many smaller
  numbers are beyond it and second when it is one
  of the numbers "beyond" whatever number we're
  using to count how many smaller numbers are
  beyond.

80
10. Permutations

• Now if we exchange elements labeled x and y how
  will the parity of the grand total change? Let's
  break up the numbers into three groups.
• The numbers either before both x and y or after
  both x and y.
• The numbers between x and y.
• x and y.

81
10. Permutations

• The same numbers will be before and after the
  numbers in group 1 so their contribution to the
  grand total will not change. The numbers in group
  2 will each contribute to the grand total twice
  once for the change in x's position (either they
  were less than x or x was less than them.) and
  again for the change in y's position. Thus their
  total contribution will be to change the grand
  total by an even number. This will not affect the
  parity of the permutation. Finally, x and y
  switching places will add or subtract 1 from the
  grand total depending on whether x or y is
  greater. Thus the parity of the grand total (odd
  or even) will change.

82
10. Permutations

• Now we are in a position to appreciate Samuel
  Loyd's famous challenge with his 14-15 puzzle
  back in the 1800's. The puzzle consisted of 15
  sliding tiles in a square frame with an empty
  space which a tile could slide into. The tiles
  were numbered from 1 to 15. The task was to get
  the tiles into the normal order

83
10. Permutations

• Sliding a tile into the empty space is equivalent
  to a transposition of the moved tile and the
  "ghost" tile in the empty space. Thus every move
  is a transposition. (read the numbers from right
  to left, top row to bottom to get the
  permutation).
• Sam Loyd offered 1000 to any one who could find
  a way to solve the puzzle from the position shown
  below.

84
10. Permutations

• Note that this is a transposition away from the
  solution. However it is impossible to solve from
  this position since it would take an odd number
  of transpositions to go from the 13-15-14
  position to the solved 13-14-15 position. But the
  empty square can only return to the lower
  right-hand corner after an even number of moves.
  The simplest way to see this is to imagine a
  checkerboard pattern of light and dark squares
  painted beneath the tiles. Then each move would
  change the color of the background square that is
  showing from light to dark or from dark to light.

85
10. Permutations

• Loyd sold a pavillion of the puzzles to suckers
  -- er, I mean puzzle fans eager to win the prize.
  He reported that
• A prize of 1,000 offered for the first correct
  solution to the problem, has never been claimed,
  although there are thousands of persons who say
  they performed the required feat. People became
  infatuated with the puzzle and ludicrous tales
  are told of shopkeepers who neglected to open
  their stores of a distinguished clergyman who
  stood under a street lamp all through a wintry
  night trying to recall the way he had performed
  the feat. The mysterious feature of the puzzle is
  that none seem to be able to remember the
  sequence of moves whereby they feel sure they
  succeeded in solving the puzzle. Pilots are said
  to have wrecked their ships, and engineers rush
  their trains past stations. A famous Baltimore
  editor tells how he went for his noon lunch and
  was discovered by his frantic staff long past
  midnight pushing little pieces of pie around on a
  plate!

86
10. Permutations

• A particularly insidious version of the puzzle
  has letters on the tiles instead of numbers. One
  of its "solved" configurations is

87
10. Permutations

• Notice that there are two 'R's and two 'A's. If
  the "wrong" 'R' and 'A' are used to begin the
  word "RATE" it will be impossible to get the
  "PAL" to come out correctly in the last row. With
  this version the group theorist "bets" the sucker
  -- er, I mean friend, that although the friend is
  accomplished at solving the 13-14-15 puzzle the
  use of letters rather than numbers will throw him
  off psychologically. The group theorist mixes up
  the tiles in front of the friend's eyes but
  leaves the 'R' in the upper left-hand corner in
  place while taking the 'A' next to it far down to
  the lower row. At the same time the 'A' in "PAL"
  is transferred near the upper left-hand corner.
  The friend is given a time limit to solve the
  puzzle. If he uses the 'R' and 'A' together that
  the group theorist has so kindly put almost into
  place for him he cuts his own throat. To solve
  the puzzle with the 'R' and 'A' as given is
  equivalent to making a transposition of the 'A's.
  As we now know, this is not possible.

88
11. Permutation Groups
89
11. Permutation Groups

• How many permutations are there on a group of n
  objects? Since there are n possible choices for
  the first position and for each of these there
  are n-1 choices for the second position and for
  each of these n(n-1) ways of choosing the first
  two there are n-2 ways of choosing the object for
  the third position etc. it turns out that there
  are n! permutations on n objects. The set of all
  permutations on n objects is called the symmetric
  group on n objects and is denoted by Sn. Since
  the factorial grows rather quickly Sn can be very
  large when n is of only moderate size. For
  example S10 contains 10! 3,628,800 elements and
  S60 contains more elements than the number of
  atoms in the universe.

90
11. Permutation Groups
91
11. Permutation Groups

• Let's look at S1. Its only element is the
  permutation
• With only one element there is only one place for
  it to go. Thus the group for S1 will have the
  following Cayley table

92
11. Permutation Groups

• Here e is the permutation that changes element 1
  to element 1. It's the only permutation possible
  and it's group under the operation followed by is
  a trivial one-element group. In fact, there is
  (up to isomorphism) only one group with one
  element. Thus we've found the structure of all
  one-element groups. (Big deal!)

93
11. Permutation Groups

• With two elements there are two (2!) possible
  permutations
• With the labels e and a their group is given by
  the Caley table e a

94
11. Permutation Groups

• This group has the identity e and a non-identity
  element a. A little reflection on the need for
  (1) an identity and (2) the cancellation rule and
  we realize that this table represents all groups
  of two elements. Thus we've "found" the structure
  of all groups of order 1 and 2. (Big deal!)

95
11. Permutation Groups

• With three elements things begin to become a bit
  more interesting. S3 has 3! 6 elements. They
  are
• Now if we perform permutation a followed by
  permutation X we end up with permutation Y.
  However if we reverse the order and perform
  permutation X followed by permutation a we end up
  with permutation Z. The group S3 is a non-abelian
  group. This calls for a short digression
  concerning notation.

96
11. Permutation Groups

• If we're using the "followed by" operation we
  need to decide on our notation. If a and b are
  two permutations does
• a b
• mean that permutation a is performed first and
  then permutation b is performed? Or does it mean
  that b is performed first and then a? Since we
  read English from left to right there is a strong
  reason to use the left-most element (in this
  case, a) as the one performed first. Thus we
  would read a b as a followed by b. However in
  mathematics the notation for operators and
  functions usually works differently. If f and g
  are two functions and I want g to act first and
  then have f act on the result of g's action then
  the usual notation for this is
• f(g(x))

97
11. Permutation Groups

• Thus the element which operates first (g) is
  written to the right of the element (f) which
  operates last. We must therefore be clear in our
  notation. Since both the leftmost-first and the
  rightmost-first notations are used in
  mathematical literature we'll have to decide
  which we'll use here. By a coin flip it has been
  determined that we will use the "followed by"
  notation where the leftmost operation is
  performed first. Thus a b will be read as a
  followed by b that is, a is performed first
  followed by performing b.

98
11. Permutation Groups

• With this convention decided upon together with
  the convention for Cayley tables that left
  element in the operation is in the column to the
  left of the table and the right element is in the
  row along the top of the table we get the
  following table for S3

99
11. Permutation Groups

• Does this table look familiar? It's exactly the
  same table as we gave for the Symmetry Group of
  the Equilateral Triangle. The two groups are
  isomorphic. This becomes clear when we realize
  that any symmetry movement of the equilateral
  triangle simply permutes the three vertices of
  the triangle.

100
11. Permutation Groups

• Exercise Write out the Caley tables for S4 and
  S5.
• Just kidding! S4 would have 576 entries in its
  Cayley table and S5 would have 14,400 entries!!

101
11. Permutation Groups

• The Symmetric Group on n objects, Sn , consists
  of all permutations on the n objects. If we limit
  ourselves to even permutations then we get an
  object called The Alternating Group on n Objects,
  An. From the name you can guess that the set of
  even permutations of n objects forms a group
  under the "followed by" operation. In fact this
  is true. The fact that e the identity permutation
  is even and that the product of two even
  permutations is an even permutation as well as
  the fact that the inverse of an even permutation
  is an even permutation makes the Alternating
  Group on n Objects, An, a subgroup of Sn.

102
11. Permutation Groups

• Theorem The order of An is half the order of Sn

103
11. Permutation Groups

• Why are permutations so important? Let's look at
  that question from another viewpoint. Let's
  define a group function. That is, a function
  whose domain is the elements of a group G and
  whose range is also the elements of G. Call this
  function fa where a is an element of the group G.
  The action of this function is to multiply any
  element of G on the right by a.
• Thus for any elements x, y or z of G we have
• fa(x) xafa(y) yafa(z) za

104
11. Permutation Groups

• For any other element, say b, of G we can also
  define fb(x) xb for all x in G. Now what are
  these functions? They are nothing other than
  permutations on the elements of G. To see this we
  only need notice that fa(x) fa(y) means that xa
  ya and by the cancellation law this means that
  x y. Also for any element c of G, c a(a-1c).
  That is, there is some element of G (namely a-1c)
  that fa sends to c. Therefore for every element a
  of G the function fa is one-to-one and onto--the
  exact definition of a permutation.

105
11. Permutation Groups

• Now we can ask ourselves, "how do these functions
  behave under composition--under the operation
  followed by?" The answer is exactly like the
  elements of G under its group operation. To see
  this notice that for any element x of G
• fa(fb(x)) fa(xb) (xb)a x(ba)
• Remember, fa(fb(x)) means that fb acts on x
  first "followed by" fa acting on the result of
  that. This is equivalent to fb fa in our
  followed-by notation.
• The repeated operation of the functions is
  equivalent to repeated multiplication on the
  right by the corresponding group element. The
  functions/permutations not only form a group
  under the operation of composition ("followed
  by") but that the group that they form is
  isomorphic to the original group G.

106
11. Permutation Groups

• This result is known as Cayley's Theorem
• Cayley's Theorem Every Group is isomorphic to a
  group of permutations.

107
11. Permutation Groups

• Cycle Notation
• There is another notation for permutations that
  shows the structure of the permutation a bit more
  clearly. It's called "cyclic notation." It works
  like this. In the permutation.
• 1 goes to 3 which goes to 4 which goes to 2 which
  goes to 5 which goes to 1 completing the cycle.
  In cyclic notation this would be written.
• Where each element is followed by the one whose
  place it goes to. The 5 at the end goes back to
  the 1 at the beginning of the cycle.

108
11. Permutation Groups

• Now consider the following permutation
• If we begin with 1 we note that 1 goes to 3 which
  goes to 7 which goes back to 1. Thus we have the
  cycle (1 3 7). Now we take one of the elements
  that isn't in the cycle just found, say, 2. 2
  goes to 5 which goes to 4 which goes to 6 which
  goes back to 2. This gives the cycle
• (2 5 4 8 6)
• Thus we have broken the permutation into two
  cycles, (1 3 7) and (2 5 4 8 6). There are a few
  things we should notice about this decomposition
  into cycles.
• The cycles are disjoint!!

109
11. Permutation Groups

• They have no element in common. This is clear
  because once we find a cycle in a permutation we
  have found where every element in the cycle goes
  under the permutation. The "destination" of every
  element of the cycle is in the cycle. Also the
  elements that have things in the cycle as their
  destination are also in the cycle. Thus once we
  close out a cycle by bringing it back to its
  first element we have found everything that the
  permutation does to the elements of that cycle
  and that they are in that cycle. If an element of
  one cycle was in another cycle then there would
  be two (at least) elements that that element was
  mapped onto by the permutation.

110
11. Permutation Groups

• So we can decompose any finite permutation into
  disjoint cycles. Just choose an element see where
  it goes and then see where that element goes and
  on until you close out the cycle. Now if there
  are any elements that are not in that cycle
  choose one of those and see where it goes etc.
  until you close out a second cycle. If there are
  any remaining elements that are not in either of
  those two cycles choose one of them and continue
  constructing cycles until you've exhausted all
  the elements of the permutation.

111
11. Permutation Groups

• Theorem Every permutation can be decomposed into
  disjoint cycles.
• Theorem Disjoint permutations commute.

112
11. Permutation Groups

• Theorem A cycle with an even number of elements
  is an odd permutation. and a cycle with an odd
  number of elements is an even permutation.
• So when we decompose a permutation into disjoint
  cycles the order of applying those cycles does
  not matter.
• Theorem If a permutation is decomposed into
  disjoint cycles such that there is an odd number
  of cycles with an even number of elements then
  the permutation is odd. If there is an even
  number of cycles with an even number of elements
  then the permutation is even.
• So once we have expressed a permutation as the
  product of disjoint cycles we can determine
  whether it is an even or odd permutation by
  noticing the number of cycles with an even number
  of elements. Since they contribute an odd number
  of transpositions to the permutation the total
  parity of transpositions, even or odd, will
  depend on the number of permutations with an even
  number of elements.

113
12. Symmetry Groups
114
12. Symmetry Groups

• The symmetric group on n letters, written Sn, is
  the group of all possible permutations on n
  letters.  The order of Sn is n!.

115
13. Dihedral Groups
116
13. Dihedral Groups

• The dihedral group Dn is the symmetry group of
  the regular n-gon (ngt2).
• Dn consists of n rotations and n reflections.
• The product of two "adjacent" reflections in Dn
  is a minimal rotation.
• Dn is generated by (a) a minimal rotation and
  any reflection, or (b) any two "adjacent"
  reflections.
• Cn is the subgroup of Dn consisting only of the
  n rotations.
• Each finite group of plane isometries is either
  Cn or Dn for some n.

117
13. Dihedral Groups

• Dihedral groups are apparent throughout art and
  nature. For example, dihedral groups are often
  the basis of decorative designs on floor tilings,
  buildings, and artwork. Chemists and
  mineralogists study dihedral groups to classify
  the structure of molecules and crystals,
  respectively. These symmetry groups are even used
  in advertising for many of the world's largest
  companies.

118
D1

• Shell Petroleum uses the symbol to the left. This
  shell shape has no rotations (other than the
  identity) and has only one mirror line
  (vertical). Therefore, like Mickey Mouse, the
  figure is said to be bilaterally symmetric and it
  fits into the category D1.

119
D2

• An example of D2 that is easily spotted is the
  logo for the Columbia Broadcasting System (CBS).
  The "eye" shape within the circle prevents the
  figure from being able to rotate by any rotation
  other than a 1/2 turn. Additionally, the figure
  has only two ways in which it can be reflected
  onto itself.

120
D3

• The luxury car, Mercedes-Benz, uses a symbol with
  three rotations and 3 mirror lines. Therefore,
  the emblem is an example of D3. If we were to
  convert this figure into a peace sign, however,
  we would lose 2 of the rotations and two of the
  reflection lines. This would leave a D1 figure.

121
D4

• The symbol for Purina is a great example of a
  finite figure of the category D4. It is easy to
  see that there are four mirror reflections of the
  figure (one vertical, one horizontal, and two
  diagonal) as well as four rotations. In other
  words, rotating the figure four times gives the
  original figure (the identity).

122
D5

• The symbol for Chrysler is a great example of a
  finite figure of the category D5. In other words,
  the symbol has five rotations and five axes of
  reflection.

123
D8

• This finite figure is a dihedral group of order 8
  due to its eight reflections and eight rotations.
  The symmetries are created by two squares placed
  on top of each other and offset by 90 degrees.

124
14. Alternating Groups
125
14. Alternating Groups

• The alternating group on n letters, written An,
  is the group of all even permutations on n
  letters.  The order of An is n!/2.  An is normal
  in Sn, in fact it is the kernel of the parity
  homomorphism.

126
14. Alternating Groups

• The group An1 defines the group of rotations of
  a generalized tetrahedron in n space, while Sn1
  defines the group of rotations and reflections. 
  This can be seen by placing any vertex in
  position, then the next, then the next, and so
  on, and reflecting if the last two must be
  swapped.

127
15. The Sylow Theorems
128
15. The Sylow Theorems

• Definitions
• A finite p-group is a group of order pn.
• Let G be a finite group of order pkr, where r is
  not divisible by p. A Sylow p-subgroup of G is a
  subgroup of order pk.

129
15. The Sylow Theorems

• The Sylow Theorems Let G be a finite group and
  let p be a prime which
• divides G. Then
• G has at least one Sylow p-subgroup.
• (ii) More generally, if p! divides G for some !
  ! 0 then G has a subgroup of order p!.
• (iii) Every p-subgroup of G is contained in a
  Sylow p-subgroup of G.
• (iv) Let P be a Sylow p-subgroup of G.
• Every Sylow p-subgroup of G is conjugate to P.
• (v) The number r of Sylow p-subgroups of G
  satisfies
• r 1 (mod p), and in addition rs
• where s is the p(-part of the order of G.

130
15. The Sylow Theorems

• LEMMA Let G be a finite p-group. Then G
  contains a subgroup of order pm whenever
• pm G.
• DEFINITION Let G be a group. A permutation
  representation of G of degree n is a
• homomorphism from G to Sn.
• LEMMA There is a bijective correspondence
  between permutation representations of
• a group G and actions of G on 1, . . . ,n.
• DEFINITION A group is said to be simple if and
  only if it has exactly two normal
• subgroups, namely e and G.

131
16. Noethers Theorem
132
16. Noethers Theorem