Abstract Algebra Prelim Aug. 2012
1. Let G1and G2be finite groups of order n1and n2. If n1and n2are relatively prime, prove
every subgroup of G1×G2has the form H1×H2, where Hiis a subgroup of Gi.
2. Let Gbe a finite group acting on a set X. For a point x∈X, prove there is a bijection between
the G-orbit of xand the left coset space G/Hx, where Hx={g∈G:gx =x}is the stabilizer
subgroup of x.
3. Let Rbe a commutative ring. The two parts of this question, about ideals in R, do not depend
on each other.
(a) Let Iand Jbe ideals in R, and let Pbe a prime ideal in R. If IJ ⊂P, prove I⊂Por
J⊂P. (Recall the product ideal IJ is the ideal in Rgenerated by all products xy with
x∈Iand y∈J.)
(b) Let P1, P2, and P3be prime ideals of R. If an ideal Isatisfies I⊂P1∪P2∪P3, then prove
I⊂Pifor some i. (Hint: Start by assuming Iis not in P1∪P2,P1∪P3, or P2∪P3.)
4. State Zorn’s lemma and then use it to prove every nonzero commutative ring contains a
maximal ideal.
5. State the class equation for a finite group Gand use it to show that every group of order 32
must have a non-trivial center.
6. Give examples as requested, with brief justification.
(a) A permutation π∈S4such that (243) = π(123)π−1.
(b) A prime ideal in Z/20Z.
(c) A unit other than ±1 in Z[√11].
(d) An integer m > 1 such that the group (Z/mZ)×is not cyclic.