Hall not implies order-isomorphic

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., Hall subgroup) need not satisfy the second subgroup property (i.e., order-isomorphic subgroup)
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Statement

Two Hall subgroups of the same order in a finite group, need not be isomorphic.

Proof

Below are some examples:

Example group Order Factorization of order Order of Hall subgroups Prime set for Hall subgroups Multiple isomorphism classes of Hall subgroups Subgroup structure information
projective special linear group:PSL(2,11) 660 2^2 \cdot 3 \cdot 5 \cdot 11 12 \{ 2,3 \} dihedral group:D12, alternating group:A4 subgroup structure of projective special linear group:PSL(2,11)
projective general linear group:PGL(2,11) 1320 2^3 \cdot 3 \cdot 5 \cdot 11 24 \{ 2,3 \} dihedral group:D24 (i.e., acting on a set of size twelve), symmetric group:S4 subgroup structure of projective general linear group:PGL(2,11)
projective special linear group:PSL(2,23) 6072 2^3 \cdot 3 \cdot 11 \cdot 23 24 \{ 2,3 \} dihedral group:D24, symmetric group:S4 subgroup structure of projective special linear group:PSL(2,23)
projective special linear group:PSL(2,59) 102660 2^2 \cdot 3 \cdot 5 \cdot 29 \cdot 59 60 \{ 2,3,5 \} alternating group:A5, dihedral group:D60 subgroup structure of projective special linear group:PSL(2,59)

GAP implementation

Here is a GAP implementation of the PSL(2,11) example. (Note that the built-in command GAP:HallSubgroup works only for solvable groups; in particular, it does not work for this group. Hence, we need to do a filtering as shown below).

gap> G := PSL(2,11);
Group([ (3,11,9,7,5)(4,12,10,8,6), (1,2,8)(3,7,9)(4,10,5)(6,12,11) ])
gap> Order(G);
660
gap> L := Filtered(List(ConjugacyClassesSubgroups(G),Representative),K-> Order(K) = 12);
[ Group([ (1,2)(3,4)(5,12)(6,11)(7,10)(8,9), (1,3)(2,4)(5,6)(7,9)(8,10)(11,12)
        , (1,5,9)(2,6,10)(3,11,8)(4,12,7) ]),
  Group([ (1,2)(3,4)(5,12)(6,11)(7,10)(8,9), (1,7,12)(2,10,5)(3,11,9)(4,6,8),
      (1,3)(2,4)(5,6)(7,9)(8,10)(11,12) ]) ]
gap> IsomorphismGroups(L[1],AlternatingGroup(4));
[ (1,2)(3,4)(5,12)(6,11)(7,10)(8,9), (1,3)(2,4)(5,6)(7,9)(8,10)(11,12),
  (1,5,9)(2,6,10)(3,11,8)(4,12,7) ] -> [ (1,4)(2,3), (1,2)(3,4), (1,2,4) ]
gap> IsomorphismGroups(L[2],DihedralGroup(12));
[ (1,2)(3,4)(5,12)(6,11)(7,10)(8,9), (1,7,12)(2,10,5)(3,11,9)(4,6,8),
  (1,3)(2,4)(5,6)(7,9)(8,10)(11,12) ] -> [ f2*f3, f3^2, f1*f3 ]
gap> IsomorphismGroups(L[1],L[2]);
fail

Here is a GAP implementation of the PSL(2,59) example.

gap> G := PSL(2,59);
<permutation group of size 102660 with 2 generators>
gap> Order(G);
102660
gap> M := Filtered(List(ConjugacyClassesSubgroups(G),Representative),K -> Order(K) = 60);
[ <permutation group of size 60 with 2 generators>, <permutation group of size 60 with 4 generators>, <permutation group of size 60 with 2 generators> ]
gap> IsomorphismGroups(M[1],AlternatingGroup(5));
[ (1,9)(2,6)(3,25)(4,10)(5,11)(7,45)(8,28)(12,48)(13,50)(14,53)(15,41)(16,30)(17,40)(18,22)(19,34)(20,59)(21,37)(23,60)(24,31)(26,35)(27,44)(29,46)(32,
    58)(33,56)(36,52)(38,47)(39,54)(42,49)(43,57)(51,55), (1,32,2)(3,33,31)(4,23,40)(5,36,26)(6,13,48)(7,10,50)(8,44,15)(9,19,39)(11,34,22)(12,37,18)(14,
    54,57)(16,51,58)(17,29,21)(20,56,49)(24,41,60)(25,45,55)(27,52,46)(28,59,38)(30,53,42)(35,47,43) ] -> [ (2,4)(3,5), (1,2,3) ]
gap> IsomorphismGroups(M[2],DihedralGroup(60));
[ (1,2)(3,4)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,
    40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33), (1,13,49,20,42)(2,52,16,45,23)(3,50,47,44,33)(4,15,18,21,32)(5,25,27,10,36)(6,22,37,46,12)(7,39,
    56,54,34)(8,51,35,41,17)(9,11,31,58,26)(14,30,24,48,57)(19,53,59,43,28)(29,60,40,38,55), (1,22,51)(2,43,14)(3,9,10)(4,56,55)(5,47,31)(6,8,42)(7,40,
    21)(11,36,50)(12,17,20)(13,37,35)(15,54,29)(16,19,24)(18,34,60)(23,59,57)(25,44,58)(26,27,33)(28,30,52)(32,39,38)(41,49,46)(45,53,48),
  (1,3)(2,4)(5,12)(6,36)(7,24)(8,11)(9,51)(10,22)(13,33)(14,56)(15,23)(16,21)(17,31)(18,45)(19,40)(20,47)(25,46)(26,35)(27,37)(28,38)(29,59)(30,39)(32,
    52)(34,48)(41,58)(42,50)(43,55)(44,49)(53,60)(54,57) ] -> [ f2*f3*f4^2, f4^2, f3^2*f4, f1*f2*f4 ]
gap> IsomorphismGroups(M[3],AlternatingGroup(5));
[ (1,13)(2,16)(3,41)(4,29)(5,47)(6,37)(7,48)(8,60)(9,57)(10,55)(11,12)(14,35)(15,52)(17,18)(19,32)(20,30)(21,27)(22,39)(23,50)(24,40)(25,58)(26,46)(28,
    43)(31,49)(33,45)(34,36)(38,56)(42,54)(44,59)(51,53), (1,3,11)(2,29,37)(4,30,48)(5,14,33)(6,47,41)(7,26,35)(8,15,27)(9,31,20)(10,36,50)(12,40,17)(13,
    25,32)(16,19,53)(18,42,43)(21,24,45)(22,55,56)(23,58,28)(34,57,51)(38,54,52)(39,59,49)(44,60,46) ] -> [ (2,4)(3,5), (1,2,3) ]
gap> IsomorphismGroups(M[1],M[2]);
fail