MATH10001 Project 2

Groups part 2

http://www.maths.manchester.ac.uk/undergraduate/ugstudies/units/2009-10/level1/MATH10001/

Subgroups

Let G be a group with binary operation .

Let H G with H .

We say that H is a subgroup of G if H is itself a group with respect to .

We write H G.

Theorem 1 (Subgroup Test)

Let G be a group with binary operation . Let H G with H . Then H is a subgroup of G if

g h-1 H for all g, h H .

Example

Let G = Z with addition and H = { 4x | x Z } = {0, 4, 8,…}.

CosetsLet H G and g G. We define the left coset

gH = {g h | h H }.

Similarly we can define the right coset

Hg = {h g | h H }.

Examples1. Let G = Z with addition and H = { 4x | x Z }.

2. Let G = the symmetry group of an equilateral triangle and H = {e, a}.

A

B C

lB

lA

lC

e = do nothing

a = reflect in line lA

b = reflect in line lB

c = reflect in line lC

r = rotate anticlockwise 120o

s = rotate anticlockwise 240o

A

C B

C

B A

B

A C

C

A B

B

C A

e a b c r s

e

a

b

c

r

s

e a b c r s

a e s r c b

b r e s a c

c s r e b a

r b c a s e

s c a b e r

Theorem 2 (Lagrange’s Theorem)

Let G be a finite group and H G. Let n be the number of left cosets of H in G. Then

|G| = |H| × n

and so |H| divides |G|.