Add new property IsRegularPGroup

[fingolfin]

Unfortunately I am not aware of an efficient algorithm for this in general. But the provided one at least performs reasonably for e.g. the groups of order 3^7 or 5^7.

[ThomasBreuer]

Very nice.
I have left a few minor comments.

Just for curiosity:
Why is Agemo not an attribute?
(Agemo is defined only for p-groups, why is it necessary to enter p as an argument?)

[fingolfin]

Agemo is not an attribute because Agemo takes 2 or 3 arguments, so it can't be an attribute. Instead, the computed values are cached in mutable attribute ComputedAgemos.

I am also puzzled why Agemo takes the prime as argument, given that p = PrimePGroup(G) must always hold (and this is enforced in Agemo). This also conflicts with the notation in the literature... But I guess we are stuck with it now.

[fingolfin]

Thanks for the careful checking and excellent feedback!

[jackschmidt]

I wrote a few more tests to improve test coverage. I didn't find any group where the last special case code applied, but the other cases all have "smallest" examples. The cases are numbered (I added debug prints to each if statement), and all but case 11 (use the definition, and get true as the answer) are very quick.

Add the following (except maybe the last two lines, case 11) to tst/testinstall/pgroups.tst right near the end

gap> IsRegularPGroup(SmallGroup(8, 3)); # Case 1
false
gap> IsRegularPGroup(SmallGroup(2187, 224)); # Case 2
false
gap> IsRegularPGroup(SmallGroup(27,3)); # Case 3
true
gap> IsRegularPGroup(SmallGroup(81, 1)); # Case 4
true
gap> IsRegularPGroup(SmallGroup(243,22)); # Case 5
true
gap> IsRegularPGroup(SmallGroup(2187, 4487)); # Case 6
true
gap> IsRegularPGroup(SmallGroup(81, 7)); # Case 7
false
gap> IsRegularPGroup(SmallGroup(78125, 684)); # Case 8
true
gap> IsRegularPGroup(SmallGroup(243, 51)); # Case 10 (no case 9 found)
false
gap> IsRegularPGroup(SmallGroup(2187, 663)); # Case 11, takes 3 seconds
true

The following would be a nice case to record, but it takes over a minute to run, so I'm not sure it is worth including.

gap> G := SmallGroup(243,22);;
gap> IsRegularPGroup(G); # Case 5
true 
gap> IsRegularPGroup(DirectProduct(G,G)); # Hup 10.3.c, Wielandt's example
false

I've run out of time myself. I wanted to find a case 9, and submit a pull request, or improve the speed on Wielandt's example, but perhaps this is enough.

[jackschmidt]

Oh, and thank you for adding such a nice check for regular p-groups :-) I'm also happy about the nice powerful p-group check above as well!