Exercise set 1

Throughout G denotes a group, and k a field.

(1)

Check that the following two definitions of a k-algebra are equivalent:

• R is a k-vector space and a ring with the underlying additive group of R in both cases being the same; the multiplication R × R → R is k-bilinear.
• R is a ring and there is a ring homomorphism k → R whose image lies in the centre of R.

(2)

Check that the following two definitions of a k-linear representation V of a group G are equivalent:

• G → GL(V ) is a group homomorphism
• V is a kG-module (in other words, there exists a k-algebra homomorphism from kG → End_k(V ))

(3)

Let V be a vector space over k of finite dimension n ≥ 1. Then End_kV is identified with the k-algebra M_n(k) of n × n matrices over k.

• For a subspace W, define ℓ_W := {φ ∈ End_kV |φ(W) = 0}. Show that ℓ_W is a left ideal in End_kV , and moreover that every left ideal of End_kV is of the form ℓ_W for some W.
• For a subspace W, define ρ_W := {φ ∈ End_kV |φ(V ) ⊆ W}. Show that ρ_W is a right ideal in End_kV , and moreover that every right ideal of End_kV is of the form ρ_W for some W.
• Show that End_kV is a simple k-algebra. (That is, it has precisely two two-sided ideals, namely, zero and itself.)

(4)

A multiplicative character of a group G is a group homormorphism from G to the (multiplicative) group k^× of the non-zero elements in k. Show that ∑ _g∈Gξ(g) = 0 for any non-trivial multiplicative character ξ of a finite group G. (A multiplicative character is called trivial if it is identically 1.)

(5)

For a finite group G determine the centre of the group algebra kG. What is its dimension as a k-vector space?

(6)

Observe that every multiplicative character of G factors through G∕(G,G). (Here (G,G) denotes the subgroup generated by the commutators (g,h) := ghg^-1h^-1, as g and h vary over all elements of G.)

(7)

Observe the following: if G is a cyclic group of order n, then the group algebra kG ≃ k[t]∕(tⁿ - 1).

(8)

Factorize the following determinant: