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 CG(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 CG(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