Definitions

Question 1

Group

Answer 1

(G, *), G non empty set
G1: ¥g,h € G, h*g € G
G2: ¥g,h,k € G, (gh)k =g(hk)
G3: exist 1_G € G, ¥g € G, 1g=g=g1
G4: ¥g€G, exist g^-1€G s.t gg^-1=1_G=g^-1g

Question 2

Subgroup

Answer 2

Suppose G is a group. H<=G means that if we restrict * to the elements of H then H is a group. H is a subgroup of G

Question 3

Subgroup Criterion

Answer 3

Suppose G is a group H c= G, then H <= G <=>

i) H =! ø
ii) ¥x,y € H, xy^-1 € H

Question 4

Right coset

Answer 4

Suppose H <= G
Let g € G
Hg = {xg | x € H}

Question 5

Odd / Even permutation

Answer 5

Let n>=2, for σ∈S_n, say σ is an even (odd) perm if σ can be written as a product of even (odd) number of transpositions

Question 6

Transposition

Answer 6

In S_n (α_1, α_2) is called a transposition

Question 7

c(σ)

Answer 7

Let n>=2, let σ ∈ S_n, c(σ) is the number of cycles when we write σ as a product of pairwise disjoint cycles.

Question 8

Alternating Group

Answer 8

Let n ∈ N, n>=2, A_n is called the set of all even permutations. A_n is called the alternating group of degree n

Question 9

C_G(S)

Answer 9

C_G(S) = {g ∈ G | gx=xg, ∀x ∈ S}

Question 10

N_G(S)

Answer 10

N_G(S) = {g ∈ G | s^g = s}

Question 11

.< s >.

Answer 11

.< s >.= {x_1,x_2,…,x_m | x_i ∈ S U S^-, m ∈ N}

Question 12

Z(G)

Answer 12

Z(G) = {g ∈ G | gx=xg ∀x∈G}

Question 13

Centraliser in G of S

Answer 13

C_G(S) = {g ∈ G | gx=xg, ∀x ∈ S}

Question 14

Centre of G

Answer 14

Z(G) = {g ∈ G | gx=xg ∀x∈G}

Question 15

Normaliser in G of S

Answer 15

N_G(S) = {g ∈ G | s^g = s}

Question 16

The subgroup of G generated by s

Answer 16

= {x_1,x_2,…,x_m | x_i ∈ S U S^-, m ∈ N}

Question 17

S^g

Answer 17

S^g = {g^-1xg | x ∈ S}

Question 18

Conjugate of S

Answer 18

S^g = {g^-1xg | x ∈ S}

Question 19

x^G

Answer 19

x^G = {g^-1xg | g ∈ G}

Question 20

Conjugacy class in G of x

Answer 20

x^G = {g^-1xg | g ∈ G} = all conjugates of x

Question 21

P-Group

Answer 21

Let p be a prime, G be a group. G is called a p-group if |G| = p^a (a ∈ N U {0})

Question 22

Cycle type of g

Answer 22

For g ∈ S_n, the cycle type of g is 1^c1, 2^c2, 3^c3…, meaning that when g is expressed as a product of pairwise disjoint cycles there are c1 1-cycles, c2 2-cycles etc etc. Convention omit i^ci if ci = 0

Question 23

Group Action

Answer 23

Suppose G is a group and Ω (!= Ø ) a set. G acts on Ω (Ω is a G set) if for each g ∈ G, each α ∈ Ω, ∃ corresponding element in Ω denoted by αg such that:
A1) ∀g1, g2 ∈ G, ∀α ∈ Ω, α(g1g2)=(αg1)g2
A2) ∀α ∈ Ω, α1 = α

Question 24

G_α

Answer 24

Suppose G is a group and Ω is a G set for α ∈ Ω, G_α = {g ∈ G| αg = α}

Question 25

Stabiliser of α in G

Answer 25

G_α = {g ∈ G| αg = α}

Question 26

Normal Subgroup

Answer 26

G a group, N <= G, N is a normal subgroup of G if Ng = gN for all g in G denoted N # G

Question 27

Simple Group

Answer 27

A group is simple means G != {1} and the only normal subgroups of G are {1} and G

Question 28

G Orbit

Answer 28

{αg | g ∈ G} = α^G = G orbit of α