Hey guys I'm struggling a bit here

1. Let G be a finite group.

a) Suppose that H is a subgroup of G. What is meant by |G : H|, the index of H in G.

b) Suppose that x ∈ G. Describe the subset of G known as the conjugacy class of x.

c) Suppose that x ∈ G. Describe the subgroup C

G(x) known as the centralizer of x.

d) State the formula which relates the size of the conjugacy class of x ∈ G to its centralizer C

G(x) .

e) Show that if the finite group G has exactly two conjugacy classes then it must be cyclic order 2.

2. Let V,W be vector spaces over the field F and let σ: V → W be a linear transformation.

a) Define the rank and nulity of σ

b) Prove that σ is injective if and only if nulity of σ = 0

c) Prove that if dimW < dimV then σ cannot be injective

Any help would be greatly appreciated! Thanks