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Math and science::Algebra::Aluffi

# 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:

$| [m]_n| = \frac{n}{\operatorname{gcd}(m, n)}$

### 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, $$|\mathbb{Z}/n\mathbb{Z}|$$.

TODO: proof
TODO: check if the properties are actually correct.

Aluffi, p48
Aluffi, p49
Aluffi, p55