THE UNIVERSITY OF BRITISH COLUMBIA
SESSIONAL EXAMINATIONS – DECEMBER 2009
MATHEMATICS 322
Time: 2 hours 30 minutes
1. [16 points] Determine whether the following statements are true or false
(you have to include proofs/counterexamples):
(a) The rings Z/35Zand Z/5Z×Z/7Zare isomorphic.
(b) The groups Z/24Zand Z/6Z×Z/4Zare isomorphic.
(c) If Gis a cyclic group of order n, and d|n, then Ghas a subgroup of
order d.
(d) The groups F∗
p2and (Z/p2Z)∗are isomorphic.
2. [13 points]
(a) Let Gbe a group, and N– a normal subgroup of G. Prove that if N
contains an element g, then Ncontains the entire conjugacy class of
g.
(b) Let σbe the following element of S4:
σ= ( 1234
2134).
Find the number of elements in the conjugacy class of σ.
(c) Prove that the permutation σfrom Part (b) cannot be contained in
any proper normal subgroup of S4.
3. [8 points] Let Mand Nbe normal subgroups of a group G. Suppose that
M∩N={e}. Prove that for every m∈Mand n∈N,mn =nm.
4. [12 points] Find the set of units and the set of zero divisors in the ring R,
where:
(a) Ris the ring Z/3Z×Z/2Z.
(b) Let Ris the quotient ring Z[√7]/I, where Iis the ideal
I={a+b√7|6|a−b}.
(Hint: find a convenient homomorphism from Z[√7] to Z/6Z).
5. [10 points] Let R=F5[x]; let I=hx2+ 1ibe the ideal in Rgenerated by
the polynomial x2+ 1, and let J=hx3+ 2ibe the ideal generated by the
polynomial x3+ 2. Prove that I+Jis a principal ideal in R, and find its
generator.
6. [10 points]
(a) Factor the element 5 ∈Z[i] as a product of irreducible elements.
(b) Is h5ia maximal ideal in Z[i]?
7. [15 points]
(a) Prove that the polynomial x3+ 2x+ 1 is irreducible in F3[x].
(b) Let Ibe the ideal I=hx3+ 2x+ 1iin F3[x], and let R=F3[x]/I . Let
α=x+I∈R. Prove that the element 1 + αhas an inverse in R.
(c) Find (1 + α)−1(that is, find γ∈R, such that γ(1 + α) = 1).
8. [8 points] Is Z[√−3] a principal ideal domain?
9. [8 points] Let Gbe a group acting on a set X. Suppose that the stabilizer
Gxof a certain point x∈Xis a proper normal subgroup of G. Prove that
every element of Gxfixes every point y∈Ox.
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