Chapter 6 Abstract algebra

1
Chapter 6 Abstract algebra

• Groups ?
• Rings ?
• Field ?
• Lattics and Boolean algebra

2

• 6.1 Operations on the set
• Definition 1An unary operation on a nonempty set
  S is an everywhere function f from S into S A
  binary operation on a nonempty set S is an
  everywhere function f from SS into S A n-ary
  operation on a nonempty set S is an everywhere
  function f from Sn into S.
• closed

3

• Associative law Let ? be a binary operation on a
  set S. a?(b?c)(a?b)?c for ?a,b,c?S
• Commutative law Let ? be a binary operation on a
  set S. a?bb?a for ?a,b?S
• Identity element Let ? be a binary operation on
  a set S. An element e of S is an identity element
  if a?ee?aa for all a ?S.
• Theorem 6.1 If ? has an identity element, then
  it is unique.

4

• Inverse element Let ? be a binary operation on a
  set S with identity element e. Let a ?S. Then b
  is an inverse of a if a?b b?a e.
• Theorem 6.2 Let ? be a binary operation on a set
  A with identity element e. If the operation is
  Associative, then inverse element of a is unique
  when a has its inverse

5

• Distributive laws Let ? and ? be two binary
  operations on nonempty S. For ?a,b,c?S,
• a?(b?c)(a?b)?(a?c), (b?c)?a(b?a)?(c?a)

Associative law commutative law Identity elements Inverse element
v v 0 -a for a
? v v 1 1/a for a?0
6

• Definition 2 An algebraic system is a nonempty
  set S in which at least one or more operations
  Q1,,Qk(k?1), are defined. We denoted by
  SQ1,,Qk.
• Z
• Z,
• N- is not an algebraic system

7

• Definition 3 Let S and T? are two
  algebraic system with a binary operation. A
  function ? from S to T is called a homomorphism
  from S to T? if ?(ab)?(a)??(b) for
  ?a,b?S.

8

• Theorem 6.3 Let ? be a homomorphism from S to
  T?. If ? is onto, then the following results
  hold.
• (1)If is Associative on S, then ? is also
  Associative on T.
• (2)If is commutative on S, then ? is also
  commutation on T
• (3)If there exist identity element e in
  S,then ?(e) is identity element of T?
• (4) Let e be identity element of S. If there
  is the inverse element a-1 of a?S, then ?(a-1) is
  the inverse element ?(a).

9

• Definition 4 Let ? be a homomorphism from S
  to T?. ? is called an isomorphism if ? is also
  one-to-one correspondence. We say that two
  algebraic systems S and T? are
  isomorphism, if there exists an isomorphic
  function. We denoted by S?T?(S?T)

10
6.2 Semigroups,monoids and groups

• 6.2.1 Semigroups, monoids
• Definition 5 A semigroup S? is a nonempty set
  together with a binary operation ? satisfying
  associative law.
• Definition 6 A monoid is a semigroup S ?
  that has an identity.

11

• Let P be the set of all nonnegative real numbers.
  Define on P by
• ab(ab)/(1a ? b)
• ProvePis a monoid.

12
6.2.2 Groups

• Definition 7 A group S ? is a monoid, and
  there exists inverse element for ?a?S.
• (1)for ?a,b,c?S,a ?(b ? c)(a ? b) ? c
• (2)?e?S,for ?a?S,a ? ee ? aa
• (3)for ?a?S, ?a-1?S, a ? a-1a-1 ? ae

13

• R-0,? is a group
• R,? is a monoid, but is not a group
• R-0,?, for ?a,b?R-0,a?bb?a
• Abelian (or commutative) group
• Definition 8 We say that a group G?is an
  Abelian (or commutative) group if a?bb?a for
  ?a,b?G.
• R-0,?,Z,R,C are Abelian (or
  commutative) group .
• Example Let G ? be a group with identity e.
  If x ? xe for ?x?G, then G ? is an Abelian
  group.

14
Example Let G1,-1,i,-i.
15

• G1,-1,i,-i, finite group
• R-0,?,Z,R,C,infinite group
• Gn is called an order of the group G
• Let G (x y) x,y?R with x ?0 , and consider
  the binary operation ? introduced by (x, y) ?
  (z,w) (xz, xw y) for (x, y), (z, w) ?G.
• Prove that (G ?) is a group.
• Is (G?) an Abelian group?

16

• R-0,? , R
• abcdef(ab)cd(ef),
• a?b?c?d?e?f?(a?b)?c?d?(e?f)?,
• Theorem 6.4 If a1,,an(n?3), are arbitrary
  elements of a semigroup, then all products of the
  elements a1,,an that can be formed by inserting
  meaningful parentheses arbitrarily are equal.

17

• a1 ? a2 ? ? an

If aiaja(i,j1,,n), then a1 ? a2 ? ?
anan? na
18

• Theorem 6.5 Let G? be a group and let
  ai?G(i1,n). Then
• (a1??an)-1an-1??a1-1

19

• Theorem 6.6 Let G? be a group and let a and b
  be elements of G. Then
• (1)acbc, implies that ab(right cancellation
  property)?
• (2)cacb, implies that ab?(left cancellation
  property)
• Sa1,,an, alai?alaj(i?j),
• Thus there can be no repeats in any row or column

20

• Theorem 6.7 Let G? be a group and let a, b,
  and c be elements of G. Then
• (1)The equation a?xb has an unique solution in
  G.
• (2)The equation y?ab has an unique solution in G.

21

• Let G? be a group. We define a0e,
• a-k(a-1)k, akaak-1(k1)
• Theorem 6.8 Let G? be a group and a ?G, m,n
  ?Z. Then
• (1)amanamn
• (2)(am)namn
• aaama,
• mana(mn)a
• n(ma)(nm)a

22

• Next Permutation groups
• Exercise P348 (Sixth) OR p333(Fifth)
  9,10,11,18,19,22,23, 24
• P355 (Sixth) OR P340(Fifth) 57,13,14,1922
• P371 (Sixth) OR P357(Fifth) 1,2,6-9,15, 20
• Prove Theorem 6.3 (2)(4)