Naming of primitive groups

[olexandr-konovalov]

Asking @hulpke and @colva - how stable are the names of primitive groups? Are we free to change them when we decide that it is sensible? While working on #34, I have seen a mix of several conventions, for example:

• S and Sym
• A and Alt
• C(N) and just N (for a cyclic group of order n).

Furthermore, we have

• [ 11, 6 ] : M(11)
• [ 55, 4 ] : M_11

Also curious why for the dihedral group there is always 2 as a factor, e.g. D(2*5) for the dihedral group of order 10 (but not for the generalised quaternion group).

[hulpke]

The names up to degree 50 are from a paper by Buekenhout. I suspect the rest is more ad-hoc. A reason for distinguishing Sn and Sym(n) could be whether it is the natural permutation representation.

Alexander

[colva]

Hi Alex

Some of the names were just made up by me at some point, and others were made up by other people. I think I never expected them to be taken very seriously, so feel free to give them a good spring clean. I did loads of htem over the course of a single afternoon, and another batch several years later.

The 2 for the dihedral group is because there are two different conventions for dihedral names, it’s to try to clarify which one

Colva

[olexandr-konovalov]

@hulpke but what about the following then:

gap> G:=PrimitiveGroup(10,8);
A(10)
gap> H:=PrimitiveGroup(45,6);
A(10)
gap> F:=PrimitiveGroup(120,18);
Alt(10)
gap> Size(G);
1814400
gap> List([G,H,F],StructureDescription);
[ "A10", "A10", "A10" ]
gap> List([G,H,F],MovedPoints);
[ [ 1 .. 10 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 
      20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 
      39, 40, 41, 42, 43, 44, 45 ], 
  [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 
      23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 
      42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 
      61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 
      80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
      99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 
      115, 116, 117, 118, 119, 120 ] ]
gap> 

PrimitiveGroup(45,6) is A(10) but it does not act on [ 1 .. 10 ].

[olexandr-konovalov]

These two groups have same name but different structure descriptions:

17^2:24 : [ [ 289, 34 ], [ 289, 35 ] ]
  [ 289, 34 ] non-isomorphic to [ 289, 35 ] : 
    (C17 x C17) : C24
    (C17 x C17) : SL(2,3)

Shouldn't it be 17^2:SL(2,3) for the 2nd?

[hulpke]

PrimitiveGroup(45,6) is A(10) but it does not act on [ 1 .. 10 ].

Names up to degree 50 are from Buekenhout and do not fit any systematic scheme.

[fingolfin]

@alex-konovalov the name for PrimitiveGroup(289, 35) is indeed misleading, there is no C_24 factor there (indeed, the set of orders of elements in this group is [ 1, 2, 3, 4, 6, 17 ]). So that's another bug in the names and should be fixed.

[olexandr-konovalov]

A fix for PrimitiveGroup(289, 35) and also for PrimitiveGroup(100,14) submitted in #36.

[olexandr-konovalov]

Next, I am going to use M(n) for Mathieu groups. We have

[ 9, 3 ] 3^2:Q(8)=M(9)
[ 10, 6 ] M(10)
[ 11, 6 ] M(11)
[ 12, 1 ] M(11)
[ 12, 2 ] M(12)
[ 21, 4 ] PSL(3, 4)=M(21)
[ 22, 1 ] M(22)
[ 22, 2 ] M(22):2
[ 23, 5 ] M(23)
[ 24, 1 ] M(24)
[ 36, 3 ] M(10)
[ 45, 2 ] M(10)
[ 1288, 1 ] M(23)
[ 1288, 2 ] M(24)
[ 1320, 1 ] M(12)
[ 1320, 2 ] M(12):2
[ 1584, 1 ] M(12):2
[ 1771, 1 ] M(23)
[ 1771, 2 ] M(24)
[ 2024, 1 ] M(24)

but also

[ 55, 4 ] M_11
[ 66, 2 ] M_11
[ 66, 3 ] M_12
[ 77, 1 ] M_22
[ 77, 2 ] M_22.2
[ 121, 48 ] M_11 wreath Sym(2)
[ 144, 3 ] M_12
[ 144, 4 ] M_12.2
[ 144, 5 ] M_12.2
[ 144, 6 ] M_11 wreath Sym(2)
[ 144, 7 ] M_12 wreath Sym(2)
[ 165, 3 ] M_11
[ 176, 3 ] M_22
[ 220, 1 ] M_12
[ 231, 1 ] M_22
[ 231, 2 ] M_22.2
[ 243, 29 ] 3^5:(2 x M_11)
[ 243, 30 ] 3^5:M_11
[ 243, 31 ] 3^5:(2 x M_11)
[ 243, 32 ] 3^5:M_11
[ 253, 4 ] M_23
[ 253, 5 ] M_23
[ 256, 150 ] 2^8:M_10
[ 276, 4 ] M_24
[ 330, 1 ] M_22
[ 330, 2 ] M_22.2
[ 396, 1 ] M_12
[ 396, 2 ] M_12.2
[ 484, 1 ] M_22 wreath Sym(2)
[ 484, 2 ] M_22^2.2^2
[ 484, 3 ] M_22^2.4
[ 484, 4 ] Aut(M_22) wreath Sym(2)
[ 495, 3 ] M_12
[ 495, 4 ] M_12.2
[ 495, 5 ] M_12
[ 495, 6 ] M_12.2
[ 506, 1 ] M_23
[ 529, 1 ] M_23 wreath Sym(2)
[ 576, 1 ] M_24 wreath Sym(2)
[ 616, 1 ] M_22
[ 616, 2 ] M_22.2
[ 672, 5 ] M_22
[ 672, 6 ] M_22.2
[ 729, 462 ] 3^6:2.M_12
[ 759, 1 ] M_24
[ 880, 1 ] M_12.2

[olexandr-konovalov]

Names up to degree 50 are from Buekenhout and do not fit any systematic scheme.

Indeed, the smallest degree where Sym and Alt appear is 55. OTOH, A and S appear across the library many times.

[olexandr-konovalov]

Mathieu groups names unification now also in #36.