Thank you for taking the time to read this and trying to help me. My question is how can I tell if a group is isomorphic to another group? This is not a homework question, I got this wrong on a past quiz and wanted to know how to solve this.
Determine which of $Z_4$ or $Z_2 \bigoplus Z_2,$ $U(8)$ and $U(10)$ are isomporphic to.
On
For your own sake, it would be a good exercise to first prove that all groups of order $4$ fall into one of two isomorphism classes: $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$.
Given this, all one must do to determine which group $U(8)$ and $U(10)$ are isomorphic to is to check for the existence of an element of order $4$. (why?)
Isomorphic groups share a lot of properties (in a sense, they share all group-theoretic properties). So, in the most general case, to show two groups are not isomorphic, it suffices to find a group invariant that they don't share. Examples of group invariants are plentiful: being Abelian, being cyclic, the number of elements of a given order, the total number of subgroups, etc.
In the particular case you are considering all the groups are of order $4$. Now, it so happens that there are only two, up to isomorphism, groups of order $4$. Do you know which groups these are? Precisely one of those is cyclic. So, all you need to do is figure out which of the subgroups you list is cyclic. That can be done very easily by trying all elements as generators.