Michigan State University Modern Algebra Worksheet

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Problem 1. Let n> 3 be an integer. Consider the dihedral group Dr.
a) Find all conjugacy classes of Dr.
(Hint: your answer should depend on the parity of n.)
b) Choose a set, R, of representatives of the conjugacy classes of D, and compute Z(r) for
each r ER
c) Prove that D. H x K for subgroups H and K of Dr with K 4D, and K = n.
Problem 2. A group generated by distinct non-identity elements ē, i, j, k satisfying (@)2 = e,
i2 = ja = k = ijk = ē is called the quaternion group Q8.
a) Prove that ē € Z(Q:).
(Hint: Show that ē commutes with ē, i, j and k)
b) It is customary to denote the identity of Q8 by 1 (instead of e) and ē by -1. For a E Qs,
define -a = (-1)a = a(-1). Prove that Q8 = {1,-1, i, -i, j, -j, k, -k}, and that |Q8= 8. Is
Q: abelian? Provide justification for your answer.
(Hint: To prove |Q8] = 8 you must show that +1, ti, Ej, Łk are distinct)
c) Show that Q: does not contain a pair of non-trivial complementary subgroups. Use the
result of Problem lc to conclude that Q8 is not isomorphic to D4.
Problem 3. Let (G,+) be an abelian group. Define a binary operation + on End(G) by
(4+4)9) = 4(9)+(9) for every 4,0 E EndG). Prove that (End(G), +) is an abelian group.
Problem 4. Let A be a finite set, with A = n.
a) Suppose that B C A. Prove that SA;B = {a E SA: a[B]C B} is a subgroup of SA. If
|B=m, prove that SA;B] = m!(n – m)!.
b) Let C = A B. Prove that SA;
BSB x Sc.
(Hint: If a E SA;B, then ab and ac defined by a = a/b, and ac = alcc
alco are in Ss and Sci
respectively)
c) Now suppose |A| = 5. Let :GⓇA + A be a faithful group action with orbits of size 3 and
2. What are the possible groups G?
(Hint: Start by using the permutation representation to show that G is isomorphic to a subgroup
of S3 x S)
Problem 5. Let G be a group of order 231 = 3 – 7.11. Let H, K and N denote sylow 3,7 and
11-subgroups of G, respectively.
a) Prove that K, N IG.
b) Prove that G HKN.
c) Prove that N