2. From Groups to Surfaces

1
2. From Groups to Surfaces
2
Groups

• Group G consists of elements, g1, g2, ... and an
  operation ². It satisfies the following
• A1. Replacing any two symbols in the equation a ²
  b c by group elements uniquely determines the
  third one.
• A2. For any three group elements gi,gj,gh we
  have (gi ² gj) ² gk gi ² (gj ² gk)

3
Usual Group Axioms

• (a ² b) ² c a ² (b ² c)
• There exists e, such that for any a
  a ² e e ² a a.
• For each a there exists a such that
  a ² a a ² a e.
• ExerciseShow that both systems of axioms are
  equivalent.

4
Finite and Infinite Groups

• The number G is called the order of group G.
• The groups of finite order are called finite
  groups. All other groups are called infinite.

5
Abelian Group

• If a ² b b ² a for any a,b 2 G the group G is
  called Abelian or commutative.

6
Residues mod n Zn.

• Two views
• Zn 0,1,..,n-1.
• Define on Z
• x y x y cn.
• Zn Z/.
• (Zn,) an abelian group, called the cyclic group
  of order n. Here is taken mod n!!!

7
Example (Z6, ).
8
Example (Z6, ) is not a group.
9
Example(Z6\0, ) is not a group.
For p prime, (Zp\0, ) forms a group.
10
R - real numbers

• (R,) is a group
• The group is Abelian and infinite. Its unit iz 0,
  the inverse of a is a.
• (R \ 0,.) is a group
• The group is infinite, Abelian, its unit is 1,
  the inverse of a is 1/a.
• Let R x 2 R x gt 0. (R,.) is also a group.
  

11
Subgroup

• H µ G is a subgroup if H is a group for the same
  group operation.
• There are subsets, closed for the group
  operation, that are not subgroups. For instance,
  (N,) is not a subgroup of (Z,).

12
Cosets

• G group
• H subgroup
• aH a ² x x 2 H
• G H t aH t bH t ...

13
Index

• Let H µ G be a subgroup of G.
• GH cosets of H is called the index of H
  in G. For finite groups GH G/H.

14
Q rational numbers

• (Q,) is a group.
• Rational numbers form a group for addition.
• (Q \ 0,.) is a group.

15
Z - integers

• (Z,) is a group.
• (Z \ 0,.) is not a group.

16
Complex numbers C.

• a a bi 2 C.
• a a bi.
• b c di 2 C.
• ab (ac bd) (bc ad)i.
• b ¹ 0, a/b (ac bd) (bc ad)i/c2 d2.
• a-1 (a bi)/(a2 b2).

17
C, - Complex numbers

• (C \ 0,.) is a group
• (C,) is a group

18
Quaternions H.

• Quaternions form a non-commutative field.
• General form
• q x y i z j w k., x,y,z,w 2 R.
• i 2 j 2 k 2 -1.
• q x y i z j w k.
• q x y i z j w k.
• q q (x x) (y y) i (z z) j (w
  w) k.
• How to define q .q ?
• i.j k, j.k i, k.i j, j.i -k, k.j -i,
  i.k -j.
• q.q (x y i z j w k)(x y i z j
  w k)

19
Quaternions H. - Exercises

• Exercise There is only one way to complete the
  definition of multiplication and respect
  distributivity!
• Exercise Represent quaternions by complex
  matrices (matrix addition and matrix
  multiplication)! Hint q a b -b a.

20
(Q,.) The Quaternion Units
21
Exercises

• N1. What is the index R\0R\0 for the
  multiplicative group.

22
Conjugation

• Given a subgroup H of a group G, then for any g 2
  G define H g-1Hg g-1hgh 2 H.
• H is called a conjugate of H.
• H is a subgroup of G.
• Conjugation is an equivalence relation on the set
  of subgroups of G.
• H H.

23
Normal Subgroup

• If H G has no nontrivial conjugates, it is
  called normal.
• For a normal group the quotient G/H forms also a
  group.
• G/H Haa 2 G
• G/H GH.

24
Group Homomorphisms and Isomorphisms

• fG1 ! G2 is a group homomorphism if for any g,h
  2 G1 we have f(g ² h) f(g) f(h), where
  (G1,²) and (G2,) are groups.
• If, in addition, f is bijection, then it is
  called an isomorphism.