Dear GAP-Forum!

On Wed, 30 Jun 1999, Malachi James wrote:

--> I am looking for some sort of command that will search a group for subgroups
--> -- possibly with certain characteristics (i.e. Search for all the subgroups
--> of some particular order of a given group).
-->
--> In particular:
--> I am wishing to see if the Alternating Group A_5 has a subgroup of order 6,
--> and
--> I want to be able to define the two non-conjugate subgroups with order 24 of
--> GL(3,2).

For finite groups G of "fairly small" order like the both mentioned
above GAP 3.4.4 is able to compute the full subgroup lattice of G. The
function which does this job is called by 'Lattice(G)' and returns a
Lattice-of-Subgroups-record wherein we can find the answers to
Malachi's questions.
By taking a closer look at the 'classes'-components of the records

Lattice( AlternatingGroup(5) )     resp.     Lattice( GL(3,2) )

we see:

(1) The Alternating Group A_5 DOES have a subgroup of order 6,
    e.g. the subgroup generated by (3,4,5) and (1,2)(4,5).
    (In fact there can be found 10 subgroups of order 6 in A_5,
     but all of them are conjugate to each other.) 
    
(2) The two non-conjugate subgroups of order 24 in GL(3,2) are
    (up to conjugacy)

G_1, generated by the four matrices
     [
      [ 0*Z(2), 0*Z(2), Z(2)^0 ],
      [ 0*Z(2), Z(2)^0, 0*Z(2) ],
      [ Z(2)^0, 0*Z(2), 0*Z(2) ]
     ],
     [
      [ 0*Z(2), 0*Z(2), Z(2)^0 ],
      [ Z(2)^0, Z(2)^0, Z(2)^0 ],
      [ Z(2)^0, 0*Z(2), 0*Z(2) ]
     ],
     [
      [ 0*Z(2), Z(2)^0, Z(2)^0 ],
      [ Z(2)^0, 0*Z(2), 0*Z(2) ],
      [ Z(2)^0, Z(2)^0, 0*Z(2) ]
     ] 
     and
     [
      [ 0*Z(2), 0*Z(2), Z(2)^0 ],
      [ Z(2)^0, Z(2)^0, 0*Z(2) ],
      [ Z(2)^0, 0*Z(2), 0*Z(2) ]
     ];

G_2, generated by the four matrices
     [
      [ 0*Z(2), 0*Z(2), Z(2)^0 ],
      [ 0*Z(2), Z(2)^0, 0*Z(2) ], 
      [ Z(2)^0, 0*Z(2), 0*Z(2) ]
     ], 
     [
      [ 0*Z(2), Z(2)^0, Z(2)^0 ],
      [ 0*Z(2), Z(2)^0, 0*Z(2) ], 
      [ Z(2)^0, Z(2)^0, 0*Z(2) ]
     ], 
     [ 
      [ 0*Z(2), 0*Z(2), Z(2)^0 ],
      [ Z(2)^0, 0*Z(2), Z(2)^0 ], 
      [ Z(2)^0, Z(2)^0, 0*Z(2) ]
     ]
     and  
     [ 
      [ 0*Z(2), 0*Z(2), Z(2)^0 ],
      [ Z(2)^0, Z(2)^0, Z(2)^0 ], 
      [ Z(2)^0, 0*Z(2), 0*Z(2) ]
     ]

Hope this helps,

Mathias

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Dipl.-Math. Mathias Kratzer           | I_nstitute for 
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