I got the following from Wikipedia:

Theorderof agroupis itscardinality, i.e., the number of its elements;

theorder, sometimesperiod, of an elementaof a group is the smallest positive integermsuch thatam=e(whereedenotes the identity element of the group, andamdenotes the product ofmcopies ofa). If no suchmexists, we say thatahas infinite order. All elements of finite groups have finite order.

Question: If I have $\displaystyle a^m$ where $\displaystyle m=4$, is it correct to say that $\displaystyle |G|=4$. If so, then $\displaystyle G $contains $\displaystyle 4$ elements. Must all the elements be distinct? Must an identity be in the group?