Download Sample Test

MATH 503: SAMPLE TEST 1
1. Sample Test 1
Problem 1 (15 points). State the definition of the kernel of a homomorphism and prove that the kernel of
any homomorphism ϕ : G → H is a subgroup of G.
Problem 2 (15 points). Suppose G acts on a set A and a ∈ A. Give the definition of the orbit of a under the
action of G.
Problem 3 (15 points). Let H = hai be a cyclic group of order n and let d be a positive divisor of n. Prove
that there is a unique subgroup of H of order d. You may use the following facts in your proof if you wish:
Proposition. With the hypotheses of the problem above, |ar | =
n
(n,r) .
Proposition. For any cyclic group hxi, |hxi| = |x|.
Problem 4 (15 points). Let D8 = {1, r, r2 , r3 , s, sr, sr2 , sr3 } be the dihedral group of order 8. Label these
elements with the numbers 1 through 8 in the order given. Define a permutation σ ∈ S8 using the action of
r3 on D8 by left multiplication, i.e. to obtain σ(n), find the x ∈ D8 corresponding to n, find the number m
between 1 and 8 inclusive which corresponds to r3 x.
Find the cycle decomposition of σ.
Problem 5 (40 points). Indicate in your bluebook whether each of the following statements is true or false.
No reasoning or proof is required for this question. However, you will lose 4 points for each incorrect answer.
(1) Every subgroup of Q8 is normal.
(2) The subgroup of rotations in D8 is its own centralizer.
(3) Every subgroup of (Q, +) is cyclic.
(4) If σ ∈ Sn is a 39-cycle, and i 6≡ 0 mod 39, then σ i is also a 39-cycle.
(5) S4 contains an element of order 8.
(6) Every finitely generated subgroup of (Q, +) is cyclic.
(7) The group (Z/32Z)× is cyclic.
(8) The group (Z/9Z)× is cyclic.
(9) The dihedral group D10 acts faithfully on the vertices of a 20-gon.
(10) There are two nonabelian groups of order 6.
Mathematics Department, Binghamton Univeristy, P. O. Box 6000, Binghamton, New York, 13902-6000
E-mail address: [email protected]
Date: February 16, 2004.
1