Groups

Question 1

What is a representation?

Answer 1

A group homomorphism f: G to GLn(R) is called a (n-dimensional) representation of G.

Question 2

First Isomorphism Theory

Answer 2

Let f: G to H be a group homomorphism. There is an isomorphism
F: G/ker f to im(f)
given by F(g(bar)) where g(bar) stands for the element (g ker f) of G/ker f.

Question 3

What is the kernel?

Answer 3

Let f: G to H be a group homomorphism. Then

ker(f) = {all g in G s.t. f(g) = e(H)}

Question 4

If f: G to H is a group homomorphism then the image f(G) is a subgroup of…

Answer 4

H

Question 5

What is a homomorphism?

Answer 5

Let G, H be groups. A map f: G to H is a group homomorphism if:
f(ab) = f(a)f(b) for all a, b in G.
If f is bijective then it is an isomorphism.

Question 6

Any group G of order p^2 is…

Answer 6

abelian.