isomorphism tests for (p-)groups

[ThomasBreuer]

Taking the problem(s) described in gap-packages/anupq/issues/32 into account, is it a good idea to delegate, in the case of p-groups G and H, from IsomorphismGroups( G, H ) to IsomorphismPGroups( G, H ), which then calls ANUPQ's functions?
Perhaps the answer is yes because other GAP methods also cannot do better if ANUPQ runs into problems.
Currently this delegation happens if ANUPQ is available and the groups have equal order larger than 2000.

Independent of that, should we perhaps introduce IsIsomorphicGroups( G, H ) for two groups G and H, returning true if G and H are isomorphic, and false otherwise?
This would be analogous to the IsIsomorphicPGroups function from the ANUPQ package, a nonconstructive test that avoids creating an isomorphism if one does not need it.
For example, if IdGroup is available for G and H then IdGroup yields a decision without creating an isomorphism, and this is often very fast.

[stertooy]

I won't weigh in on the first question, but regarding the second, I'm very much in favour of adding an IsIsomorphicGroups function.

In addition to the advantages for speed/efficiency, it's also more intuitive to (new) users. The question whether such a function exists has popped up on e.g. math.stackexchange before. And searching for this through e.g. Google may lead users to believe it does exist, because there are still GAP 3 manuals online that mention the function IsIsomorphicGroup.

Also good to know, is that the SONATA packages defines IsIsomorphicGroup which already does this. Though (personally) I'd rather have this as core GAP functionality.

[ThomasBreuer]

Thanks for the hint about SONATA.

It defines IsIsomorphicGroup as an operation with seven methods, all with the same requirements (IsGroup for both arguments) and each with an individual incremental rank, in order to call the methods in the prescribed ordering.
From my point of view, the only advantage of this setup could be that one is able to insert additional methods between the given ones, but I think this is hypothetical.

I would install just one method in the GAP library, which is ranked above the SONATA methods and which checks the criteria in question. (The IdGroup test should better be based on ID_AVAILABLE not on an explicit list of group orders.)

[hulpke]

Do we want to worry about the naive user calling if IsIsomorphic(G,H) then iso:=IsomorphismGroups(G,H); and thus replacing one isomorphism test with two ? That was part of my reason for not introducing IsIsomorphic when I added IsomorphismGroups.

[limakzi]

I am very much in favour of adding IsIsomorphicGroups ranked above SONATA, especially for $p$-groups. My personal case: I load sonata mostly only for this function.

How about IsIsomorphicGroups calls iso := IsomorphismGroups(G,H) and returns true if iso fails?

[ThomasBreuer]

Do we want to worry about the naive user calling if IsIsomorphic(G,H) then iso:=IsomorphismGroups(G,H); and thus replacing one isomorphism test with two ? That was part of my reason for not introducing IsIsomorphic when I added IsomorphismGroups.

In the documentation of IsomorphismGroups, we can recommend calling IsIsomorphicGroup if only the flag is needed.
In the documentation of IsIsomorphicGroup, we can recommend calling IsomorphismGroups if this isomorphism object will be needed later on if it exists.
Logically, it should be clear that GAP does not have caches for the results, which would admit to reuse a "known" IsIsomorphicGroup result in an IsomorphismGroups call.

How about IsIsomorphicGroups calls iso := IsomorphismGroups(G,H) and returns true if iso fails?

Currently this is what IsIsomorphicGroup does when cheaper criteria do not yield an answer.

(Concerning the function name:
ANUPQ introduces IsIsomorphicPGroup and IsPqIsomorphicPGroup, SONATA introduces IsIsomorphicGroup.
This suggests that the function name should be IsIsomorphicGroup not IsIsomorphicGroups?)

[limakzi]

This suggests that the function name should be IsIsomorphicGroup not IsIsomorphicGroups?)

OK. 👌

[hulpke]

(Concerning the function name: ANUPQ introduces IsIsomorphicPGroup and IsPqIsomorphicPGroup, SONATA introduces IsIsomorphicGroup. This suggests that the function name should be IsIsomorphicGroup not IsIsomorphicGroups?)

This brings to mind CRLG's favorite anecdote about the student's answer to the exam question "Are A and B isomorphic" ? (Answer: A is , B is not)

[ThomasBreuer]

(Concerning the function name: ANUPQ introduces IsIsomorphicPGroup and IsPqIsomorphicPGroup, SONATA introduces IsIsomorphicGroup. This suggests that the function name should be IsIsomorphicGroup not IsIsomorphicGroups?)

This brings to mind CRLG's favorite anecdote about the student's answer to the exam question "Are A and B isomorphic" ? (Answer: A is , B is not)

Well, one can argue that the name IsIsomorphicGroups should better be replaced by AreIsomorphicGroups, whereas IsIsomorphicGroup( G, H ) can be read as "Is G isomorphic with H?" and therefore makes some sense.
(It would be a bad solution to introduce both names, and the worst case would be if these two functions would compute different results.)

Thus I keep the name proposal IsIsomorphicGroup, as in SONATA.