MATH 6320 QUAL, AUGUST 2025

Let \(p\) be an odd prime. Prove that there are only two non isomorphic groups of order \(2p\).

Let \(\mathbb {Q}\) be the set of rational numbers, considered a group with respect to the usual addition. Prove that \(\mathbb {Q}\) is indecomposable, that is, it is not isomorphic to a direct sum \(A \oplus B\) of two non-trivial subgroups.

Let \(\zeta _8 = \exp (2\pi i/8)\), complex eight root of one. Use Galois theory to determine all subfields of \(\mathbb {Q}(\zeta _8)\).

Let \(F\) be a field of characteristic \(p\). Let \(S_n\) be the group of permutations of \(X = \{1,2,\ldots ,n\}\). Let \(V = F[X]\) be the space of \(F\)-valued functions. It is a linear representation of \(S_n\), where \(\sigma \in S_n\) acts on \(f \in F[X]\cong F^n\) by \((\pi (\sigma )f)(x)=f(\sigma ^{-1}x)\). We have an intertwining map \(P:F[X]\to F\), where on \(F\) we have the trivial action of \(S_n\), given by \(f\mapsto f(1)+f(2)+\cdots +f(n)\). Prove that this map splits, that is, there exists an intertwining map \(Q\) in the opposite direction such that \(P\circ Q=1\), if and only if \(p\) does not divide \(n\).