Abstract Algebra Prelim Jan. 2008
1. Let Gbe a group of order 21.
a) Use the Sylow theorems to show Ghas a normal subgroup of order 7 and to determine how
many subgroups it could have of order 3.
b) If Ghas a normal subgroup of order 3, show Gis abelian.
c) Use semidirect products to construct an example of a nonabelian group of order 21.
2. Let Iand Jbe ideals in a commutative ring Rsuch that I+J= (1).
a) Define multiplication of ideals and show IJ =I∩J.
b) Show R/(I∩J)∼
=R/I ×R/J as commutative rings. (Part a is not needed.)
3. a) Show Z[i] is a PID.
b) Show any nonzero prime ideal in a PID is a maximal ideal.
4. Let Hand Kbe subgroups of G, and let Gact on the set G/H ×G/K in a diagonal manner:
g(aH, bK) = (g aH, gbK ). (We don’t assume Hor Kis normal, so G/H and G/K are just
sets, not necessarily groups.)
a) Show the stabilizer subgroup of (aH, bK)∈G/H ×G/K is aH a−1∩bKb−1.
b) If Hand Khave finite index in G, use the action of Gon G/H ×G/K to show there is a
normal subgroup NCGof finite index contained in H∩K.
5. State Zorn’s lemma, then state a theorem whose proof uses Zorn’s lemma, and then give the
proof of that theorem. (Be sure to verify the conditions of Zorn’s lemma are satisfied within
the proof).
6. Give examples of
a) a character of order 4 on the additive group Z/12Z(either give a formula or a table of
values). Recall a character of a finite abelian group is a homomorphism from the group to C×.
b) a solvable group Gwhere G06={e}and G00 ={e}, with a computation of G0. (The notation
G0is the commutator subgroup of G, and G00 = (G0)0.)
c) a maximal ideal in Z[X] which contains the ideal (X2+ 1).
d) a unit other than ±1 in Z[√10], along with its inverse.