ALGEBRA PRELIM September, 1997

Let \(G\) be a finite group with the property that for each positive integer \(n\), there are at most 2 subgroups of order \(n\) in \(G\). Prove that \(G\) is cyclic.

Let \(V\) be a finite dimensional vector space over an algebraically closed field \(F\). Let \(T\) be a linear transformation of \(V\) into itself, let \(\lambda \) be an eigenvalue of multiplicity \(m\) of \(T\) (i.e., \(m\) is the multiplicity of \(\lambda \) as a root of the characteristic polynomial of \(T\)), and let \(U\) be the subspace \(U=\{v\in V\mid T(v)=\lambda v\}\). Prove that the dimension of \(U\) is \(\le m\).

Let \(R\) be a ring and let \(L\) be the set of all nilpotent elements of \(R\).

1. If \(R\) is commutative, prove that \(L\) is an ideal.
2. Give an example of a ring \(R\) in which \(L\) is not an ideal.

Prove that there is a polynomial \(p(x)\) with rational coefficients such that the Galois group of \(p(x)\) over the field of rational numbers is cyclic of order 5.

Let \(H\) be a proper subgroup of the finite group \(G\). Prove that \(G\) has a conjugacy class \(C\) such that \(C\) contains no element of \(H\).

Let \(A\) and \(B\) be finite rings with the same number of elements. Assume that \(n\) is an integer \(>1\) and \(x^n=x\) for all \(x\in A\) and \(y^n=y\) for all \(y\in B\).

1. If \(n=6\), prove that \(A\) and \(B\) are isomorphic.
2. If \(n=4\), prove by an example that \(A\) and \(B\) need not be isomorphic.

Let \(E\) be an extension of the field \(F\) and let \(\alpha \) be an element of \(E\) which is algebraic over \(F\) of degree \(n\).

1. If \(n\) is prime to 2, prove that \(F(\alpha ^2)=F(\alpha )\).
2. Give an example in which \(n\) is prime to 3 but \(F(\alpha ^3)\ne F(\alpha )\).

Let \(A_n\) be the alternating group of degree \(n\) (i.e., \(n\) is the number of objects permuted by \(A_n\)). If \(n\ge 5\) and \(H\) is a subgroup of \(A_n\) such that the order of \(H\) is \(\frac {(n-1)!}{2}\), then prove that \(H\) is isomorphic to \(A_{n-1}\).

Let \(R\) be a semisimple right Artinian ring (in particular, this implies that \(R\) has an identity 1). Assume \(x\in R\) and define the sets \(A_x\) and \(B_x\) by \[ A_x=\{y\in R\mid y\ne 0\text { and }xy=0\} \quad \text {and}\quad B_x=\{y\in R\mid xy=1\}. \] Prove that exactly one of these sets is empty.

Let \(F\) be a field such that the multiplicative group of \(F\) is cyclic. Prove that \(F\) is a finite field.