# Find the order of every element in the symmetry group of the square $D_4$

So I know this group has 8 elements. But I am not sure how to interpret the 'order' of those elements. Order in a cyclic group is the smallest positive integer $n$ s.t. $a^n=e$. But I thought dihedral groups weren't cyclic.

Any help would be appreciated!

Dihedral groups aren't cyclic, but for any finite group $G$ and any $g \in G$, $g^{|G|} = e$, where $|G|$ denotes the number of elements in the group. The order of an element $g \in G$ is the unique smallest natural number $d$ such that $g^d = e$ and $g^f \neq e$ for any $1 \leq f < d$. Moreover, the order $d$ of an element divides the order of the group. Thus, for your question, you should only find orders of $d = 1, 2, 4$ or $8$ -- actually, not $8$, because as you pointed out $D_4$ is not cyclic.