Abstract Algebra Prelim Jan. 2012
1. (a) Define a p-Sylow subgroup of a finite group.
(b) For each prime p, prove that any two p-Sylow subgroups of a finite group are conjugate.
(That is, prove the second part of the Sylow theorems.)
2. Let the additive group Zact on the additive group Z[1
3] = {a/3k:a∈Z, k ≥0}by ϕn(r)=3nr
for n∈Zand r∈Z[1
3]. Set G=Z[1
3]oϕZ, a semi-direct product.
(a) Compute the product (r, m)(s, n) and the inverse (r, m)−1in the group G.
(b) Show Gis generated by (1,0) and (0,1).
3. Let Rbe a ring with identity, possibly noncommutative. Let Iand Jbe two-sided ideals in
R. Define IJ to be the set of finite sums a1b1+···+anbn=Pn
k=1 akbkwhere n≥1, ak∈I,
and bk∈J.
(a) Prove that IJ is a two-sided ideal in Rand that IJ ⊂I∩J.
(b) If Ris commutative and I+J=Rthen prove IJ =I∩J, indicating where you use the
commutativity in your proof.
(c) Let R= ( Z Z
0Z) = {(a b
0c) : a, b, c ∈Z}, which is a noncommutative ring under addition
and multiplication of matrices. Set
I= ( 0Z
0Z) = {(0y
0z) : y, z ∈Z}and J= ( Z Z
0 0 ) = {(x y
0 0 ) : x, y ∈Z}.
Show Iand Jare two-sided ideals in R,I+J=R, and IJ 6=I∩J. (This shows that
part b becomes false in general if we drop its commutativity hypothesis.)
4. (a) Show the only units in Z[√−5] are ±1.
(b) Define what it means for an integral domain Rto be a unique factorization domain
(UFD) and use the equation 2 ·3 = (1 +√−5)(1−√−5) to show Z[√−5] is not a unique
factorization domain.
5. Let Rbe a commutative ring. Show a nonzero ideal Iin Ris a free R-module if and only I
is a principal ideal with a generator that is not a zero divisor in R. (Hint: For the direction
(⇒), show a basis of Ican’t have more than one term in it.)
6. Give examples as requested, with brief justification.
(a) A group action which has no fixed points.
(b) The class equation for a non-abelian group that is not isomorphic to S3. (Be sure to
specify what the group is.)
(c) A cyclic R[X]-module that is three-dimensional as a vector space over R.
(d) A unique factorization domain (UFD) which is not a principal ideal domain (PID).