Are there general procedures to find group isomorphisms? This question is actually a series of related questions that have been bothering me while studying group theory. It seems to me that if we have two groups $G,H$ of same order and $G$ has a subgroup of order $n$ and $H$ has no subgroup of order $n$, then we can't have an isomorphism.

*

*Now, suppose that we have two subgroups $G,H$ and they have the same quantity of subgroups $G_1,\dots,G_n$ and $H_1,\dots,H_n$ and $|G_i|=|H_i|$ does this means we have an isomorphism? Or are there groups where these conditions are true but there is no isomorphism?


*I've been thinking about mappings from generating sets of groups, if we can map each generator of $G$ to another generator of $H$ such that each generator from $H$ and $G$ have the same order, do we have an isomorphism? Perhaps this is equivalent to the previous one but I am not really sure.
This is bothering me because in my lectures, we speak about what is an isomorphism but we never go in depth to actually talk about general procedures to find isomorphisms. Perhaps it's too early? I don't know.
 A: Determining whether two (finite) groups are isomorphic is quite difficult in general. There are lots of obstacles to isomorphism - basically, any non-silly property you can think of is preserved by isomorphisms, so any non-silly disagreement between two groups witnesses their non-isomorphicness - but it's difficult to guarantee an isomorphism without explicitly constructing one. (In particular, per the comments above neither of your conditions is sufficient to conclude that the two groups are isomorphic.)
That said, there are some tools we can use.

*

*In the abelian case, group isomorphism is actually easy. The fundamental theorem of finite abelian groups gives a canonical way to describe any finite abelian group such that two finite abelian groups are isomorphic iff they have the same description. And the analysis required to get that description isn't very hard.


*Even in the non-abelian case, we can show that a putative isomorphism between two groups $G$ and $H$ must satisfy certain constraints, and so narrow the range of possibilities we have to consider. For example, it has to send Sylow subgroups to Sylow subgroups appropriately, so breaking down the two groups via Sylow theory can make it easier to find an isomorphism if one exists (or prove that there is no isomorphism, for that matter).
