Preliminary Examination in Algebra September 1994

1. No group of order 156 is simple.
2. Every group of order 289 is abelian.

If a square matrix \(M\) of complex numbers satisfies the equation \[ M^{1994}=\text {Identity}, \] then \(M\) is diagonalizable.

1. Define the elementary divisors of a square matrix \(M\) over the principal ideal domain \(R\).
2. Calculate the elementary divisors of the matrix of integers \[ \begin {pmatrix} 4 & 6\\ 6 & 12 \end {pmatrix}. \]

1. Let \(R\) be a ring and let \(M\) be an \(R\)-module. Prove that if \(M\) is generated by simple submodules, then \(M\) is the direct sum of simple submodules.
2. Show that every vector space over a field has a basis.

Show that if the field \(F\) contains a primitive \(n^{\mathrm {th}}\)-root of unity, then the Galois group of the polynomial \(x^n-a\in F[x]\), \(a\in F\), is abelian.

Show that if \[ 0\to K\to P\to M\to 0 \] and \[ 0\to K'\to P'\to M\to 0 \] are short exact sequences of \(R\)-modules, where \(P\) and \(P'\) are projective, then \[ K\oplus P'\cong K'\oplus P. \]

If \(R\) is a ring and \(M\) is a simple \(R\)-module, then \(\operatorname {End}_R(M)\) is a division ring.

If \(R\) is a left noetherian ring, then \(R[x]\) is left noetherian.

If \(p(x)\in F[x]\) where \(F\) is a finite field, then \(p(x)\) is separable over \(F[x]\).

A finite subgroup of the multiplicative group of nonzero elements of a field is cyclic.

Show that the elements \(e_1\wedge e_2+e_3\wedge e_4\), \(e_1\wedge e_3+e_3\wedge e_4\), and \(e_5\wedge e_6+e_1\wedge e_3\) are linearly independent in \(\Lambda ^2\mathbb {R}^6\), where \(\{e_1,e_2,e_3,e_4,e_5,e_6\}\) is a basis for \(\mathbb {R}^6\).

Explain what a free group is and prove that any two free bases for the free group \(F\) have the same cardinality.

Prove that the group given by the presentation \[ \langle x,y\mid x^2=1,\ y^2=1,\ xyx=yxy\rangle \] has 6 elements.

1. Explain what it means for a contravariant functor \(F\colon \mathcal {C}\to \mathbf {Set}\) to be representable (‘Set’ is the category of sets).
2. Show how the power set \(P(S)=\{T\mid T\subseteq S\}\) defines a contravariant functor \(P\colon \mathbf {Set}\to \mathbf {Set}\) and prove that \(P\) is representable.