Theorems

Question 1

Lagranges Theorem

Answer 1

If G is a finite group and H <= G then |G| = |H|[G:H] where [G:H] is the number of right cosets in G a.k.a index of H in G

Question 2

|GL_n(q)|

Answer 2

|GL_n(q)| = (q^n-q^0)(q^n-q^1)…(q^n-q^n-1)

Question 3

Class equation of a finite group

Answer 3

Suppose G is a finite group. Let x1,..,xk ∈ G. One chosen from the conjugacy class of G. Set n_i = |xi^G|

i) |G| = SUM_1^k n_i = SUM_1^k [G: C_G(xi)]
ii) |G| = |Z(G)| + SUM_l+1^k ni
iii) |G| = |Z(G)| + SUM_l+1^k [G: C_G(xi)]

Question 4

Cauchy Theorem

Answer 4

Let p be a prime and G a finite group. If p | |G| then G has at least one element of order p

Question 5

Classification theorem for finitely generated abelian groups

Answer 5

Any finitely generated abelian group is isomorphic to a direct product of cyclic groups G ~= Zm1 x Zm2 x … x Zmk x Z^s, for some s > 0 and m1|m2….mk-1|mk
s = rank of G
m1,m2,m3… are the torsion coefficients of G

Question 6

G a group and N <= G. Then N # G is equivalent to any of the following:

Answer 6

i) G=N_G(N)
ii) the conjugates of N = {N}
iii) ∀g ∈ G, ∀n ∈ N, g^-1ng ∈ N
iv) for every g in G, Ng=gN, all right cosets of N are left cosets

Question 7

First Isomorphism Theorem

Answer 7

Suppose A:G –> K is a homomorphism and G and K are groups. Then G/kerA ~= AG

Question 8

Sylows Theorem

Answer 8

Suppose G is a finite group. |G| = p^r m where p is a prime, r in Z, p!|m. Then,

i) ∃at least 1 subgroup of P of G with |P| = p^r
ii) subgroups of G order p^r they all conjugate
iii) if x <= G and x is a p-group then x<=P^g for some g in G
iv) if n is the number of subgroups of G of order p^r then n|m and n=1modp

Question 9

For any z in x^G then…

Answer 9

z^G = x^G

Question 10

Two permutations in Sn are conjugate <==>

Answer 10

They have the same cycle type

Question 11

Two permutations in Sn have the same cycle type <==>

Answer 11

They are conjugate

Question 12

If g belongs to C_G(S) then…

Answer 12

S^g=S and therefore g belongs to N_G(S)

Question 13

If S <= N_G(S) then…

Answer 13

|S| | |N_G(S)|

Question 14

G is a disjoint union of its…

Answer 14

Conjugacy classes

Question 15

|Omega| |G_a| =

Answer 15

|G|

Question 16

Suppose G is a group, H <= G and [G:H] = 2 then

Answer 16

H is a normal subgroup of G

Question 17

If |G| = pq where p and q are distinct primes such that p !| q-1 then…

Answer 17

G is cyclic and G has a normal Sylow p-group

Question 18

First iso theorem

Answer 18

Suppose ø:G->K is a homomorphism with G and K group then G/kerø ~= Gø

Question 19

Jordan holder

Answer 19

Suppose G is finite group but not trivial group
*) G $ H1 …. $ Hr ={1}
**) G $ K1 … $ Ks ={1}
Then i) r=s
ii) the simple group {G/H1, …, Hr-1/Hr} and same for K are the same up to multiplicity and isomorphism