S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 2: Group Theory Basics

2.6 The order of an element and its properties

We’ve introduced order; now we develop its theory more fully.

Proposition 2.6.1 (Powers and order). Let have finite order . Then:

  1. .
  2. .
  3. The elements are all distinct.

Proof (Properties of order). (1) () If , then . () Write with . Then . By minimality of , , so .

(2) by (1).

(3) If for , then with , contradicting minimality of . ∎

Proposition 2.6.2 (Order of powers). Let . Then .

Proof (Order of powers). Let . We show .

First, , so .

Conversely, let . Then , so . Write and where . Then , so . Since , we have , i.e., . Thus . ∎

Corollary 2.6.1 (Generators of cyclic groups). In of order , the element is a generator iff .

Example 2.6.1. In , the generators are (the four elements coprime to 12). The element has order .

Question 4.

  1. In a group , if and , what are the possible orders of ?
  2. Prove: if and with and , then .
  3. Find the order of in .
Show solution

(1) divides both 12 and 8, so . Possible orders: 1, 2, 4.

(2) Let . Then (using commutativity), so .

Now (since ), so . Since , . Similarly, . Since , . Thus .

(3) .