\(
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\require{physics}
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\)
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:
[\[ | [m]_n| = \frac{?}{\quad ? \quad } \] ]
Property 4
Following from property 3:
- The class \( [m]_n \) generates \( \mathbb{Z}/nZ \) iff [what?].
- the order of every element of \(
\mathbb{Z}/n\mathbb{Z} \) [what can be said here?].
