1
$\begingroup$

Let $K$ be a field, $G$ a group and $G'=[G,G]$ the commutator subgroup of $G$. Show

  1. Two matrix representations of $G$ over $K$ of degree $1$ are equivalent only if they are identical.
  2. The group $G$ and the factor group $G/G'$ have the same number of matrix representations over $K$ of degree $1$ .
  • 0
    Yes you are right , the second part has to be restricted to degree 1 .2012-04-15

1 Answers 1

2

Hint #1: $GL_1(K)\cong K^*$ is commutative.

Hint #2: If $\rho: G\to K^*$ is a group homomorphism, what can you say about $\rho(G')$ in light of the first hint?

Hint #3: Let $p:G\to G/G'$ be the projection homomorphism. Show that if $\rho_1'$ and $\rho_2'$ are two distinct representations of $G/G'$, both of degree 1, then $\rho_1=\rho_1'\circ p$ and $\rho_2=\rho_2'\circ p$ are two distinct representations of $G$, both of degree 1.

Hint #4: Show that Hint #2 implies that if $\rho$ is any representation of $G$ of degree 1, then there exists a degree 1 representation $\rho'$ of $G/G'$ such that $\rho=\rho'\circ p$.

  • 0
    @Vedananda: A couple more hints :-)2012-04-17