224 CHAPTER 4 Abstract Algebra
(c) Show that for every element x ∈ U, x^3 = e, where e is the
identity ofU.
(d) Show thatU isnotabelian.- Let (G,∗) be a group such that |G| ≤4. Prove thatG must be
abelian. - Let (G,∗) be a group such that for alla, b∈G we have (ab)^2 =
a^2 b^2. Prove thatGmust be Abelian. Find elementsa, b∈Sym(X)
in Exercise 3 above such that (ab)^26 =a^2 b^2. - Let Abe a set. Show that (2A,+) is an Abelian group, but that
if|A|≥2 then (2A,∩) and (2A,∪) arenotgroups at all. - Here’s another proof of the fact that if p is prime, then every
element a ∈ Z∗p has an inverse. Given that a, p are relatively
prime, then by the Euclidean trick (page 58) there exist integerss
andtwithsa+tp= 1. Now what?
4.2.5 Cyclic groups
At the end of the previous subsection we observed that the multiplica-
tive group (Z∗ 7 ,·) has every element representable as a power of the
element 3. This is a very special property, which we formalize as fol-
lows.
Definition of Cyclic Group.Let (G,∗) be a group. If there exists an
elementg∈G such that every element is a power (possibly negative)
ofx, then (G,∗) is called acyclic group, and the elementxis called
ageneratorofG. Note that a cyclic group is necessarily Abelian. To
see this, assume that the groupGis cyclic with generatorxand that
g, g′∈G. Theng=xmandg′=xn for suitable powersm, n, and so
gg′ = xm∗xn = xm+n = xn+m = xnxm = g′g,