Examples of groups

Question

If B is a groups and b is in B, what are some examples where |b| is greater than or equal to |b^2| and |b|=|b^2|? And also how did you figure this out?

Answer 1

In $(\mathbb Z / 5 \mathbb Z, +)$ the order of $2$ is equal to the order of $4$, i.e. $5$. That is a general fact that in a group whose order is equal to a prime number $p$, the order of all elements is equal to $p$ except for the identity element.

But in $(\mathbb Z / 12 \mathbb Z, +)$, the order of $2$ is $6$ while the order of $4$ is $3$.