Preliminary Examination in Algebra
Department of Mathematics & Statistics
Directions: There are 8 questions, all of the same weight. Please take the time to ensure
accuracy and completeness, especially for the questions you find easiest.
not mean excessive verbosity. You should not attempt to prove standard propositions that you
cite except where the proof of a standard proposition is explicitly sought.)
The ring of integers will be denoted by Z and its field of fractions by Q.
1. Prove that a non-abelian group of order 2p, p an odd prime, must have a trivial center.
2. When F is a field, let GLn(F ) denote the group of all invertible n × n matrices in F
under the operation of matrix multiplication, and let SLn(F ) denote its subgroup defined
by restricting to matrices of determinant 1. Find a subgroup H of GLn(F ) such that
GLn(F ) is isomorphic to the semi-direct product of H with SLn(F ).
3. Prove that the number of elements in any finite field must be a prime power.
4. Let Z/mZ denote the ring of integers modulo m. Let r, s be positive integers.
(a) What element of Z generates the ideal rZ+ sZ?
(b) What is the kernel of the canonical ring homomorphism Z/rsZ −→ Z/rZ× Z/sZ?
(c) Find an integer t such that Z/rZ⊗ Z/sZ ∼= Z/tZ.
5. Let M be a 3× 3 matrix over the rational field Q whose characteristic polynomial is
t3 + 2t2 − 4t− 8
(a) all possible sequences of (polynomial) invariant factors for M .
(b) representatives of the different possible similarity classes of such matrices M .
6. For any integer n ≥ 3 let Dn denote the nth dihedral group, i.e., the group of order 2n
that is the semi-direct product of the cylic group Z/nZ with Z/2Z for the unique non-
trivial action (by automorphisms) of the latter on the former, or, equivalently, the group
of symmetries of a regular n-gon.
(a) Describe Dn by generators and relations.
(b) Show that every automorphism of the dihedral group D3 is inner, i.e., is the conju-
gation by some element of D3.
(c) Show that for any n odd, n ≥ 5, t