Let H be a subgroup of the finite group G, and let p be a
prime. Prove that two distinct Sylow p-subgroups of H cannot lie
in the same Sylow p-subgroup of G.
Let n be a positive integer and let R be a ring with a 1.
Show that R has characteristic n if and only if R has a subring
(with the same 1) isomorphic to (i.e. the integers modulo
n).
Let A and B be normal subgroups of the group G.
Prove that is a normal subgroup of G.
Prove that is isomorphic to a subgroup of .
If G is finite and ,
prove that AB= G.
Let G be a simple group with a subgroup H of index 6.
Prove that there exists a momomorphism .
Prove there are no simple groups of order 300.
Let R be the ring
(so ). Define
by for .
Show that for , .
Let . Show that is a unit if and only if
.
Show that has no solution with .
Show that and are
irreducible elements of R.
Deduce that R is not a UFD.
Let R be a UFD and let S be a multiplicatively closed
subset of nonzero elements of R.
If u is irreducible in R, prove that u is either
irreducible or a unit in S^{-1}R.
Prove that S^{-1}R is a UFD.
Let V be a vector space over and let be
a linear transformation. Describe how V can be made into an
-module via T.
Suppose V has basis (e_{1},e_{2},e_{3}) and T is the linear
transformation defined by T(e_{1}) = 2e_{1}, T(e_{2}) = -4e_{2}-4e_{3}, and
T(e_{3}) = 9e_{2} + 8e_{3}.
Express V as a direct sum of two nonzero -modules.
Calculate (x^{2} - 4x +4)e_{2}.
If where f_{1} | f_{2} |
... | f_{n} and f_{1} is not a unit, what are the possibilities for
the ideals (f_{i})?
Express V as a direct sum of cyclic modules.
Does there exist such that ?
Let R be a PID, let p be a prime of R, and let M be
the R-module where the
e_{i} and n are positive integers. Define and .
Prove that M(p) and pM are submodules of M.
Prove that .
In the case and p=x^{2}+1, give an example of
a finitely generated R-module M such that .
Let and ,
where is the ring of integers and . Prove that , where is the field of rational numbers.
List without repetition all the abelian groups of order 3^{2}
2^{3}. Which ones are cyclic?