PRELIMINARY EXAMINATION IN ALGEBRA

Show that for any positive integer \(n\), every element of order 2 in the alternating group \(A_n\) is the square of an element of order 4 in the symmetric group \(S_n\).

Let \(G\) be a finite \(p\)-group, with \(|G|>p\). Prove that the order of \(\operatorname {Aut}(G)\) is divisible by \(p\).

Let \(R\) be a ring with 1. A left \(R\)-module \(M\) is called simple if \(M\ne 0\) and if the only submodules of \(M\) are \(M\) and 0. Show that every simple module is isomorphic to \(R/I\) for some maximal left ideal \(I\) and that \(I\) is unique if \(R\) is commutative.

In the category of \(\mathbb {Z}\)-modules, is the module \(\mathbb {Q}/\mathbb {Z}\) (a) projective? (b) injective? (c) flat?

Justify your answer.

Let \(G\) be a group of order \(p^2q\), where \(p\) and \(q\) are distinct primes. Show that \(G\) has a normal Sylow subgroup.

Let \(M\) be a 5 by 5 matrix with real coefficients such that \(M^2=2M-I\). Show that the subspace of \(\mathbb {R}^5\) consisting of vectors fixed by \(M\) has dimension at least 3.

Let \(R\) be a commutative ring with 1. Show that every \(R\)-module is free if and only if \(R\) is a field.

Compute the number of monic irreducible polynomials of degree 3 over the field \(\mathbb {Z}_7\).

Let \(F\) be a field that contains a primitive \(n\)th root of unity. Show that if \(a\) is an element of \(F\) and the field \(E\) is obtained from \(F\) by adjoining an \(n\)th root of \(a\), then \(E\) is a Galois extension of \(F\) with cyclic Galois group.

State and prove Hilbert’s basis theorem.