# Hall not implies order-isomorphic

From Groupprops

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) neednotsatisfy 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

An example is the group . This group has order , and the -Hall subgroups have order 12. It turns out that there are two possible isomorphism classes of such subgroups: the dihedral group of order twelve (i.e., acting on a set of size six), and the the alternating group on four elements.

Closely related are the following examples:

- The group has order , and the -Hall subgroups have order 24. It turns out that there are two possible isomorphism classes of such subgroups: the dihedral group of order 24 (i.e., acting on a set of size twelve), and the symmetric group on four elements.
- The group has order , which has prime factorization . The -Hall subgroups of this are in three isomorphism classes: the alternating group, the dihedral group, and the cyclic group of order .
- The group has , with prime factorization . The -Hall subgroups of this group fall in two isomorphism classes: the alternating group on five letters, and the dihedral group of order .

## GAP implementation

Here is a GAP implementation of the 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 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