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Math and science::Algebra::Aluffi

Groups. Order.

There are two related concepts of order: the order of an element, and the order of a group.

Element order

Let \( g \) be an element of group \( G \). \( g \) is said to have finite order iff there exists a natural number \( n \) such that \( g^n = e \). We say that the order of \( g \) is \( k \) iff \( k \) is the smallest natural such that \( g^k = e \). We write \( |g| = k \).

If there is no such natural number, \( g \) does not have finite order and we write \( |g| = \infty \).

Notation: \( e \) is shorthand for \( e_G \), the identity element of group \( G \). \( g^n \) is notation for \( g \bullet g \bullet ... g \), \( g \) composed with itself \( n \) times.

Group order

Let \( G \) be a group. If \( G \) has finite elements, then we say that the order of \( G \) is the number of its elements. We write \( |G| \) to denote the order of \( G \). Otherwise, when \( G \) has infinite elements, we write \( |G| = \infty \).

4 Lemmas

For a group \( G \) and element \( g \), \( |g| \le |G| \).

Proof. This is vacuously true if \( |G| = \infty \). If \( G \) has finite order, then consider \( |G| + 1 \) powers of \( g \): \( g^0, g^1, g^2 ... g^{|G|} \). All of these powers can't be unique, otherwise \( G \) would have more than \( |G| \) elements. Let \( g^i = g^j \) be two repeated elements in the sequence (with \( i < j \)). Then \( g^{j-i} \) must be the identity, and we have \( |g| \le j - i \le |G| \).

Let \( g \) be an element of a group. If \( g^n = e \), then \( |g| \) is a divisor of \( n \).

Aluffi cheats a little in his proof. I think the proof requires an inductive proof. I think it would be worth checking out how the inductive proof is set up.

An immediate consequence of the above lemma is the following corollary:

Let \( g \) be an element with finite order, and let \( N \in \mathbb{R} \). Then:

\[ g^N = e \iff N \text{ is a multiple of } |g|.\]

Let \( G \) be a group and \( g \in G \) be an element of finite order. Then for any \( m > 0 \), \( g^m \) has finite order. Specifically, the order of \( g^m \) is related to the order of \( g \) as follows:

\[ |g^m| = \frac{\operatorname{lcm}(m, |g|)}{m} = \frac{|g|}{\operatorname{gcd}(m, |g|)}. \]

Source

p47