1Groups and homomorphisms

IA Groups

1.3 Cyclic groups

The simplest class of groups is cyclic groups. A cyclic group is a group of the

form

{e, a, a

2

, a

2

, · · · , a

n−1

}

, where

a

n

=

e

. For example, if we consider the

group of all rotations of a triangle, and write

r

= rotation by 120

◦

, the elements

will be {e, r, r

2

} with r

3

= e.

Officially, we define a cyclic group as follows:

Definition (Cyclic group C

n

). A group G is cyclic if

(∃a)(∀b)(∃n ∈ Z) b = a

n

,

i.e. every element is some power of a. Such an a is called a generator of G.

We write C

n

for the cyclic group of order n.

Example.

(i) Z is cyclic with generator 1 or −1. It is the infinite cyclic group.

(ii) ({+1, −1}, ×) is cyclic with generator −1.

(iii) (Z

n

, +) is cyclic with all numbers coprime with n as generators.

Notation.

Given a group

G

and

a ∈ G

, we write

hai

for the cyclic group

generated by

a

, i.e. the subgroup of all powers of

a

. It is the smallest subgroup

containing a.

Definition

(Order of element)

.

The order of an element

a

is the smallest integer

n

such that

a

n

=

e

. If

n

doesn’t exist,

a

has infinite order. Write

ord

(

a

) for the

order of a.

We have given two different meanings to the word “order”. One is the order

of a group and the other is the order of an element. Since mathematicians

are usually (but not always) sensible, the name wouldn’t be used twice if they

weren’t related. In fact, we have

Lemma. For a in g, ord(a) = |hai|.

Proof.

If

ord

(

a

) =

∞

,

a

n

6

=

a

m

for all

n 6

=

m

. Otherwise

a

m−n

=

e

. Thus

|hai| = ∞ = ord(a).

Otherwise, suppose

ord

(

a

) =

k

. Thus

a

k

=

e

. We now claim that

hai

=

{e, a

1

, a

2

, · · · a

k−1

}

. Note that

hai

does not contain higher powers of

a

as

a

k

=

e

and higher powers will loop back to existing elements. There are also no repeating

elements in the list provided since a

m

= a

n

⇒ a

m−n

= e. So done.

It is trivial to show that

Proposition. Cyclic groups are abelian.

Definition

(Exponent of group)

.

The exponent of a group

G

is the smallest

integer n such that a

n

= e for all a ∈ G.