228 CHAPTER 4 Abstract Algebra
- Let G be a finite group of order n, and assume that G has a
elementgof ordern. Show thatGis a cyclic group and thatgis
a generator. - Let G be a finite cyclic group of order n and assume that x is
a generator of G. Show that |G| =o(x), i.e., the order of the
groupGis the same as theorder of the elementx. - Let G be a cyclic group of order n. Show that the number of
generators of G is φ(n). (Hint: let x∈ G be a fixed generator;
therefore, any element of G is of the form xk for some integer
k, 0 ≤k≤n−1. Show thatxk is also a generator if and only if
k andnare relatively prime.)
4.2.6 Subgroups
Most important groups actually appear as “subgroups” of larger groups;
we shall try to get a glimpse of how such a relationship can be exploited.
Definition. Let (G,∗) be a group and letH ⊆ G be a subset ofG.
We say thatHis asubgroupofGif
(i)H is closed under the operation∗, and(ii) (H,∗) is also a group.Interestingly enough, the condition (i) above (closure) is almost
enough to guarantee that a subset H ⊆ G is actually a subgroup.
There are two very useful and simple criteria each of which guarantee
that a given subset is actually a subgroup.
Proposition. Let(G,∗) be a group and let H ⊆ Gbe a non-empty
subset.
(a) If for any pair of elements h, h′ ∈ H, h−^1 h′ ∈ H, then H is a
subgroup ofG.