MATH 6320 QUAL, MAY 2025

Prove that there are only two non isomorphic groups of order 6.

Let \(n\) and \(m\) be two relatively prime natural numbers. Let \(N = nm\). Let \((G,+)\) be an abelian group such that \(0 = Ng = g+\cdots +g\) (\(N\) summands) for every \(g \in G\). Let \(G_n=\{g\in G \mid ng=0\}\) and \(G_m=\{g\in G \mid mg=0\}\). Then \(G_n\) and \(G_m\) are subgroups of \(G\). (You do not need to verify that.) Prove that \(G\) is isomorphic to \(G_n \oplus G_m\).

Let \(F\) and \(E = F(\alpha )\) be two fields such that \([E:F]\) is odd. Prove that \(E = F(\alpha ^2)\).

Let \(G\) be a finite group. Let \(V_1\) and \(V_2\) be two non-isomorphic irreducible complex representations. Let \(V = V_1 \oplus V_2\). Compute the algebra of linear transformations of \(V\) intertwining the action of \(G\): \[ \operatorname {Hom}_G(V,V). \]