ALGEBRA PRELIM EXAM August 22, 2006

Let \(F\) be a field. Let \(X\) denote the set of 8-by-8 matrices \(A\) with entries in \(F\) such that \(A^3=0\) but \(A^2\ne 0\). Observe that \(G=\operatorname {GL}(8,F)\) acts on \(X\) by conjugation. Compute the number of orbits for the action of \(G\) on \(X\).

Let \(R\) be a commutative integral domain with identity. Recall that an \(R\)-module is torsion free if for any \(r\in R\) and \(m\in M\), \(rm=0\) implies that either \(r=0\) or \(m=0\).

1. Suppose \(R\) is a principal ideal domain and \(M\) and \(N\) are two \(R\) modules. Prove that if \(M\) and \(N\) are torsion free, so is \(M\otimes _R N\).
2. Give an example of an integral domain \(R\) and two torsion free \(R\)-modules \(M\) and \(N\) such that \(M\otimes _R N\) is not torsion free.

Describe the structure of a Sylow 2-subgroup of the symmetric group \(S_n\) in terms of products and semidirect products of known groups for:

1. \(n=4\); and
2. \(n=8\).

Let \(G\) be a finite group and let \(\mathbb {C}[G]\) denote the complex group algebra of \(G\). Show that the invariants functor which (on the level of objects) takes a \(\mathbb {C}[G]\) module \(M\) to \[ M^G:=\{m\in M\mid f\cdot m=m\text { for all }f\in \mathbb {C}[G]\} \] is an exact functor.

Suppose \(R\) is a commutative ring with identity. Let \(J\) denote the intersection of all maximal ideals in \(R\). Suppose \(M\) is a finitely-generated \(R\)-module such that \[ JM:=\{jm\mid j\in J\text { and }m\in M\}=\{0\}. \] Prove that \(M=\{0\}\).

Determine the Galois group of \(x^4-2\) over each of the following fields: the complex numbers; the real numbers; the rational numbers; the integers modulo 5.

Determine the group of units in the ring \(\mathbb {Q}[x]/((x^2+1)^2)\).

Determine (up to isomorphism) all groups \(G\) which satisfy the following condition: there exists a proper subgroup \(H\) of \(G\) which contains all proper subgroups of \(G\).

Let \(V\) be a finite-dimensional vector space over a field of characteristic \(p>0\). Let \(T:V\longrightarrow V\) be a linear transformation with \(T^{p^n}=\operatorname {Id}\) for some \(n\). Prove that there is a nonzero vector \(v\in V\) with \(Tv=v\).

Prove that there are no simple groups of order 992.