In general, what techniques can be used to show that 2 groups are not isomorphic? Say I have 2 groups $G$ and $H$, what techniques can be used to show that they are not isomorphic?
A simple one I can think of is proving that their order is different, thus showing there cannot be a bijection in between the 2. However I am interested in other approaches as well.

Look for any difference in the groups such as


*

*Order of the groups.

*One group has an element of order $n$, and the other does not have an element of order $n$.

*One group has a subgroup of order $n$, and the other does not.

*The orders of the centers of the groups are different.

*One group is Abelian, the other is not.

*One group is cyclic, the other is not.

