TITLE: In general, what techniques can be used to show that 2 groups are not isomorphic?
QUESTION [13 upvotes]: 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.

REPLY [17 votes]: The Group Isomorphism problem for finitely presentable groups is undecidable. Restricting to finite groups, the problem is certainly decidable, with the trivial algorithm of enumerating all $n!$ permutations (assuming of course, the two groups have the same order). 
If the input groups $G, H$ are given by their Cayley tables (which is a generous assumption), we can improve the bound to $n^{O(\log(n))}$ in the following way. First, observe that for a group $G$ of order $n$, there exists a generating set of size $\log_{2}(G)$. To see this, we consider the following algorithm: 
-Set $K := \emptyset$. 
-While $\langle K \rangle \neq G$, select $g \in G \setminus \langle K \rangle$, and add $g$ to $K$.
Each new element we add to $K$ at least doubles the size of $\langle K \rangle$. So $|K| = \log_{2}(n)$. 
As $\langle K \rangle = G$, we seek to map the elements of $K$ into a generating set of $H$. There are $|H|^{|K|} = n^{\log(n)}$ such functions. Accounting for some overhead in testing such functions, we achieve the $n^{O(\log(n))}$ bound. This bound has been relatively unscathed, though several classes of groups have polynomial time algorithms for the isomorphism problem. Some of these classes include abelian groups (https://www.sciencedirect.com/science/article/pii/S0022000007000293), groups without abelian normal subgroups (https://link.springer.com/chapter/10.1007%2F978-3-642-31594-7_5), and groups with abelian Sylow towers (http://drops.dagstuhl.de/opus/volltexte/2012/3400/).