Groups. Some properties of order.

Property 1

Suppose that \( g^2 = e \) for all elements \( g \) in group \( G \). Then we can say that \( G \) is commutative.

Property 2

Suppose \( g \) is an element with odd order. Then \( |g^2| = |g| \)?

Property 3

The order of \( [m]_n \) in \( \mathbb{Z}/n\mathbb{Z} \) is 1 if \( n | m \), and more generally:

Property 4

Following from property 4:

• The class \( [m]_n \) generates \( \mathbb{Z}/nZ \) iff \( \operatorname{gcd}(m, n) = 1 \).
• the order of every element of \( \mathbb{Z}/n\mathbb{Z} \) divides \( n \), which is the order, \( |\mathbb{Z}/n\mathbb{Z}| \).