Definitions and theorems

Question 1

What are the 4 things needed to be a group?

Answer 1

G1 ∀ a,b belonging to G, a*b belongs to G CLOSED UNDER BINARY OPERATION
G2 ∀ a,b,c in G, (ab)c=a(bc) ASSOCIATIVITY
G3 there is an element 1G in G st, ∀ a in G, 1a=a1=a IDENTITY
G4 ∀ g in G, there is an element g^-1 st gg^-1=g^-1g INVERSE

Question 2

what is the subgroup criterion

Answer 2

h=/= empty set
∀ a,b, ab is in H
∀ a in H, a^-1 is in H

2nd and 3rd can be combined st its sufficient to prove ab^-1 remains in H

Question 3

what is lagrange theorem

Answer 3

if H ≤ G then |G|=[G:H] |H|. [G:H] is the index of H in G.

In particular, |H| divides |G|

Question 4

what is centralizer of S in G

Answer 4

CG(S)={g in G|gx=xg for all x in S}

Question 5

what is normalizer of S in G

Answer 5

NG(S)={g in G| S^g = S}

Question 6

what is subgroup of G generated by S

Answer 6

={x1,….,xm | m in N, xi belongs to SUS^-}

Question 7

what is a conjugate in G of S

Answer 7

S^g={g^-1 x g | x belongs to S}

Question 8

H K | = ?

Answer 8

|H| . |K| / | H ∩ K |

Question 9

what does it mean if G acts on Ω

Answer 9

αg is a unique element st

(A1) g1,g2 in G, α in Ω, then α(g1g2)=(αg1)g2
(A2) for all α in Ω, α1=α

Question 10

G orbit is…………………….?

Answer 10

α^G={αg | g belongs to G}

Question 11

Stabilizer is…………..?

Answer 11

Gα={g in G | αg=α}

Question 12

Class equation of a finite group

Answer 12

|G| = Σni = Σ[G:CG(xi)]
|G| = |Z(G)| + Σni
|G| = |Z(G)| + Σ[G:CG(xi)]

Question 13

Right coset of a in G is?

Answer 13

Ha={ha | h∈H}

Question 14

[G:H], the index of H in G is defined as what?

Answer 14

The number of right cosets of H in G. T

Question 15

GLn(F) = …

Answer 15

Set of nxn matrices A with entries in F, such that det(A)=/=0

Question 16

GLn(F) | = …

Answer 16

(q^n - 1)(q^n - q)(q^n - q^2)…(q^n - q^n-1)

Question 17

What is a transposition?

Answer 17

A cycle of length 2

Question 18

Every permutation in Sn can be written as what?

Answer 18

A product of transpositions

Question 19

σ is an even permutation if …

Answer 19

σ is an even permutation if σ can be written as a product of an even number of transpositions (same for odd)

Question 20

If σ can be written as a product of r1 transpositions and also a product of r2 transpositions, then what can you say about r1 and r2

Answer 20

either both even or both odd

Question 21

An = ….

Answer 21

An = {σ |σ is an even permutation in Sn}

Question 22

A3 = ….

Answer 22

A3 = {(1), (123), (132)}

Question 23

For n ≥ 2, An is a subgroup of …

Answer 23

An ≤ Sn