[GAP Forum] hall-senior number vs small group id

[Vipul Naik]

Hello,

I just double-checked the Hall-Senior list provided by Eamonn O'Brien
against the GAP 3 catalogue. The only discrepancies are with the
abelian groups, probably because abelian groups were treated as a
separate layer and hence did not follow the Hall-Senior numbering
procedure.

Hence, for non-abelian groups (which is most groups!), the code
provided by Bettina Eick should suffice.

For groups of order 16, there are two discrepancies:

gap> F := Filtered([1..14], i -> not(SmallGroupToHallSenior[4][i] = Gap3CatalogueIdGroup(SmallGroup(16,i))[2]));
[ 2, 5 ]
gap> List(F,i -> StructureDescription(SmallGroup(16,i)));
[ "C4 x C4", "C8 x C2" ]

For groups of order 32, there are two discrepancies:

gap> F := Filtered([1..51], i -> not(SmallGroupToHallSenior[5][i] = Gap3CatalogueIdGroup(SmallGroup(32,i))[2]));
[ 3, 16, 21, 36 ]
gap> List(F,i -> StructureDescription(SmallGroup(32,i)));
[ "C8 x C4", "C16 x C2", "C4 x C4 x C2", "C8 x C2 x C2" ]

For groups of order 64, there are two discrepancies:

gap> F := Filtered([1..267], i -> not(SmallGroupToHallSenior[6][i] = Gap3CatalogueIdGroup(SmallGroup(64,i))[2]));
[ 2, 50, 55, 183, 192, 246 ]
gap> List(F,i -> StructureDescription(SmallGroup(64,i)));
[ "C8 x C8", "C32 x C2", "C4 x C4 x C4", "C16 x C2 x C2", "C4 x C4 x C2 x C2", "C8 x C2 x C2 x C2" ]

Vipul

* Quoting Vipul Naik who at 2012-04-17 22:39:54+0000 (Tue) wrote
> There is a slight discrepancy between the GAP 3 catalogue numbers and
> those defined in Hall-Senior's book.
> 
> For order 16, there is a mix-up between nummbers 3 and 4:
> 
> According to Hall-Senior:
> 
> (16,3) corresponds to Z4 X Z4 (GAP ID (16,2)) and (16,4) corresponds
> to Z8 X Z2 (GAP ID (16,5))
> 
> According to the GAP 3 catalogue:
> 
> (16,4) corresponds to Z4 X Z4 (GAP ID (16,2)) and (16,3) corresponds
> to Z8 X Z2 (GAP ID (16,5))
> 
> Vipul
> 
> * Quoting Bettina Eick who at 2012-04-17 08:59:58+0000 (Tue) wrote
> > 
> > Dear Forum,
> > 
> > as far as I remember, one can get this information as follows.
> > 
> > The groups of order 64 as determined by Hall and Senior are part
> > of the solvable groups library of GAP 3. The id of a group in this
> > GAP 3 catalogue can be obtained in GAP 4 via Gap3CatalogueIdGroup.
> > Thus with
> > 
> > gap> hs := List(AllSmallGroups(64), x -> Gap3CatalogueIdGroup(x)[2]);;
> > gap> Position(hs, 187);
> > 245
> > 
> > one finds that the Hall-Senior group 187 is SmallGroup(64,245);
> > 
> > Best wishes,
> > 
> > Bettina
> > 
> > 
> > 
> > On Mon, 16 Apr 2012, sumeyra uskudar wrote:
> > 
> > >dear forum,
> > >is there a function in gap (or anywhere) which can give the small group id
> > >of a group when we give the hall-senior number? or visaversa?
> > >
> > >Indeed, I need the subgroup structure of the group of order 64 which has
> > >Hall Senior number 187. but I can not get the small group ID from anywhere
> > >thus it is hard to define it in GAP.
> > >
> > >-- 
> > >*Sümeyra Bedir*
> > >_______________________________________________
> > >Forum mailing list
> > >Forum at mail.gap-system.org
> > >http://mail.gap-system.org/mailman/listinfo/forum
> > >
> 
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