HOMEWORK ASSIGNMENT 1
DUE: TUESDAY, JANUARY 15
The first week overlaps the group theory covered in Introduction to Abstract Algebra. It is
quite likely you have seen some of these problems before.
(1) An element g of a group G is called an involution if g2 = 1.
(a) How many involutions are there in Sn?
(b) Let G be a group such that every element is an involution. Prove that G is
abelian.
(2) (a) If ß is an n-cycle, prove that ßk is a product of gcd(n, k) disjoint cycles, each of
length n/ gcd(n, k).
(b) If p is a prime, then prove that every power of a p-cycle is either a p-cycle or 1.
(c) A permutation a ? Sn is regular if either a = 1 or a has no fixed points and is
the product of disjoint cycles of the same length. Prove that a is regular if and
only if there exists an n-cycle ß and a positive integer m such that a = ßm. (Hint
for the “only if” part: If a = (a1 . . . ak)(b1 . . . bk) · · · (z1 . . . zk), a product of m
disjoint k-cycles and n = mk, then let ß = (a1b1 . . . z1a2b2 . . . z2 . . . akbk · · · zk).)
(3) Show than an r-cycle is an even permutation if and only if r is odd.
(4) Give an example of a group and elements in that group to show that the relation “x
commutes with y” is not transitive.
(5) (a) A permutation matrix P over a field F is an n×n matrix obtained from permuting
the columns of the n×n identity matrix I (over F). In other words, if the columns
of I are I = [ e1 e2 . . . en ], then P = [ ea1 ea2 . . . ean ] for some a ? Sn. Prove
that the set of all n × n permutation matrices is a subgroup of GL(n,K) which
is isomorphic to Sn.
(b) Prove that every finite group is isomorphic to a group of matrices.
(6) Let G be a group and fix a ? G. Define ?a : G ? G by ?a(x) = axa-1. (This
mapping is called conjugation by a.)
(a) Prove that ?a is an automorphism.
(b) Define G : G ? Aut(G) by G(a) = ?a for all a ? G. Prove that G is a homomorphism.
(7) Prove that a group G is abelian if and only if the map x ?? x-1 is an automorphism.
(Since the mapping is obviously a bijection, it is only the property of being
a homomorphism which is important here.)