Missing methods for IsCentral

[olexandr-konovalov]

I've got an example from a user:

gap> M:=DihedralGroup(16);
<pc group of size 16 with 4 generators>
gap> e:=Elements(M)[4];
f3
gap> IsCentral(M,e);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `IsCentral' on 2 arguments called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
called from read-eval loop at line 23 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>

Indeed, the manual says that in IsCentral( M, obj ) the 2nd argument must either be an element or a magma, but there seem no methods installed for this case at all:

brk> ShowMethods();
#I  Searching Method for IsCentral with 2 arguments:
#I  Total: 9 entries
#I  Method 1: ``IsCentral: for two associative FLMLORs-with-one'', value: 86
#I  Method 2: ``IsCentral: for two associative FLMLORs'', value: 76
#I  Method 3: ``IsCentral: for two associative rings-with-one'', value: 70
#I  Method 4: ``IsCentral: generic method for two groups'', value: 70
#I  Method 5: ``IsCentral: for two FLMLORs'', value: 68
#I  Method 6: ``IsCentral: for two associative rings'', value: 60
#I  Method 7: ``IsCentral: for two magmas-with-inverses'', value: 34
#I  Method 8: ``IsCentral: for two magmas-with-one'', value: 28
#I  Method 9: ``IsCentral: for two magmas'', value: 22

I've advised to use e in Centre(M) instead, but I think that a missing method should be added as well. Moreover, Centre is defined as an attribute for magmas.

[stevelinton]

Sounds sensible. Do we need to worry about deciding whether we want Centre or LieCentre?

[fingolfin]

We are not taking about just one missing method here, though; rather, we'd potentially need one for each of the cases listed by ShowMethods above. Note that all but one of the existing IsCentral methods are implemented using IsCentralFromGenerators (the one exceptions is for groups).
So, we could add a companion to IsCentralFromGenerators, say IsCentralElementFromGenerators, and then use that to add the missing methods (plus one extra method for groups.

(Actually, we could probably use IsCentralFromGenerators and IsCentralElementFromGenerators for groups, too. I am not sure why this is not the case already -- I'd expect Comm( u, g ) <> one (which is used in the group implementation) to be slower than g1 * g2 <> g2 * g1 in general (except perhaps for IsPcGroup and IsPcpGroup, but those should then have their own optimized version of these functions anyway).