Prove that a group of order p^{n}, where p is a prime and , has a nontrivial center.
Let G be a group of order pq, where p and q are primes
and p < q. Prove that G has only one subgroup of
orderq.
Show that if H is a subgroup of order 12 in a group G of
order 36, then H is a normal subgroup of G. (Map G to the
automorphisms of the set of right cosets of H.)
NOTE: there are
examples
of groups of order 36 with subgroups of order
12 which are not normal.
Describe all abelian groups of order 36, up to
isomorphism.
Prove that if p(x) is a polynomial irreducible over a field
F, and if and are roots of p(x) in some
extension field of F, then and are
isomorphic. What happens if p(x) is reducible?
Let L be a normal extension of a field F, let
, and let H be a subgroup of G. Prove that
if for all , then K
is a subfield of L;
if K is normal over F, then H is a normal subgroup of
G.
Let R be a commutative principal ideal domain (PID) with a 1.
If , show that a and b have a greatest common divisor (i.e. d divides a and b, and if g divides a and b,
then g divides d).
Show that is not a PID.
Let R be a commutative ring with identity, and let I,J be
ideals in R. Let for
.
Show that IJ is an ideal of R.
Show that if I+J = R, where , then (recall that R has a 1).
Suppose further that R is a domain, and that
for all ideals I,J in R. Prove that R is a field. (Take
principal ideals.)