PH. D. QUALIFYING EXAM FALL 2008 - ALGEBRA
Answer all of the questions. Here Cndenotes the cyclic group of order n.
1. Classify all normal subgroups of the symmetric group S5.
2. Show that any group of order 3393 is not simple.
3. Let F⊆K⊆Lbe groups. Prove that [L:F] = [L:K][K:F].
4. Let f=x6+ 1 ∈Q[x]. Factor fover Qand let gbe the unique factor of highest
degree. Find the Galois group of g.
5. Find the rational canonical forms for all 3 ×3 matrices Awith rational entries that
satisfy A4=I3.
6. Prove that for any polynomial f∈F[x], where Fis a field, there is a splitting field
for f.
7. For (a) G=C4and (b) G=C2×C2let V=CGbe the complex group algebra of G
viewed as a CG-module. Explicitly write Vas a direct sum of irreducible CG-sub-modules.
8. Show that a Euclidean domain is a principal ideal domain.
9. Prove Gauss’s Lemma (for the integers): if f∈Z[x] is reducible over Q, then fis
reducible in Z[x].
10. Let Fbe a free group on two generators a, b. Let Nbe a subgroup of index 4 in F.
Then Facts on the four right cosets of Nin Fby permuting them (for c∈Fand any coset
Ng, the action of con the cosets is N g 7→ Ngc), thus giving a finite permutation group P.
What are the possibilities for P?
Typeset by A
M
S-T
E
X
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