By an ``N-group", we mean a finite group with the property
that every nonidentity homomorphic image has a nonidentity center.
Prove that maximal subgroups of N-groups are always normal.
Let R be an integral domain and let K be its field of
fractions. Assume that if , then either or
. Prove that
R is a local ring.
R is integrally closed in K.
Let R be a PID and let A, M be nonzero finitely generated
R-modules.
Show that if A is torsion free, then .
Provide a counterexample to the conclusion of (1)
in the case A is not torsion free.
Assume that p and q are distinct primes. Show that a group
of order p^{2}q cannot be simple.
Let k be a field. If , define
Prove that if f_{1},f_{2}, ... is a countable list of polynomials in
k[X_{1}, ... ,X_{n}], then there is a positive integer T such that
Let k be a field. Prove that if A and B are two -matrices with entries in k, both of which have minimal polynomial
X^{n-1}, then A and B are similar.