Algebra Prelim, August 2016
Do all problems
- Let G be a simple group of order
240 = 24·3·5.
- Prove that G has a subgroup of order 15, and that all groups of
order 15 are cyclic.
- Prove that G has exactly 32 elements of order 3. (Hint: show 15
| NG(P)| where P is a Sylow 3-subgroup.)
- Let R be a UFD with the property that any ideal that can be
generated by two elements is principal.
I1⊆I2⊆... is an ascending
chain of principal ideals in R, prove that there exists
N∈ℕ such that In = IN for all n > N.
- Prove that R is a PID.
- Let R be a ring with a 1 and let S be a subring of R with the
same 1. Prove or give a counterexample to the following statements.
- If P is a projective left S-module, then
R⊗SP is a
projective left R-module.
- If P is an injective left S-module, then
R⊗SP is an
injective left R-module.
- Let R be a PID and let M and N be finitely generated
R-modules. Suppose that
M3≌N2. Prove that there exists
an R-module P such that
- Let k be an algebraically closed field of characteristic 2 and let
A be a square matrix over k such that A is similar to A2.
- Prove that A is similar to A2n for all positive integers n.
- Prove that A is similar to a diagonal matrix over k.
- Explicitly construct a subfield K of
ℂ such that
[K : ℚ] = 3 and K is Galois over
ℚ. For such a
field K, prove that
K(√2) is a Galois extension of
ℚ, and determine the Galois group
- Let H be a central subgroup of the finite group G and let
be a character of H. Assume that
H≠G. Prove that
IndHG(χ) (the induced character) is not an irreducible
character of G (all characters are assumed to be over