TITLE: Find a subgroup of index $3$ in $\mathbb{Z}^*_{13}$
QUESTION [5 upvotes]: Find a subgroup of index $3$ in $\mathbb{Z}^*_{13}$. I know that the subgroup must be of order 4, and that the identity has to be in it. Also, every element in the subgroup must have an inverse in the subgroup.

REPLY [2 votes]: Subgroup generated by 8 (or 5) seems to have four elements in it H={[1],[5],[8],[12]}. Identity is there, 12 is self-inverse, 5 and 8 are inverses of each other.