I want to calculate the number of all orders of elements in HS (Higman-Sims sporadic simple group). Is there any way of doing this with MAGMA or GAP? How I can determine orders of elements of a group with CharacterTable
in GAP?
1 Answer
In general, having a group in GAP, information about the orders of its elements could be derived from its conjugacy classes and orders of their representatives, for example:
gap> G:=Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ]);
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ])
gap> List(ConjugacyClasses(G),Size);
[ 1, 56, 24, 24, 21, 42 ]
gap> List(ConjugacyClasses(G),c -> Order(Representative(c)));
[ 1, 3, 7, 7, 2, 4 ]
Clearly, for a large group having this information precomputed would be extremely useful, and due to the Atlas of Finite Group Representations, such opportunity exists. For example, you may see the information about conjugacy classes in the Atlas page on $HS$ here without using any computational algebra system.
The GAP Character Table Library derives some data from Atlas, and you may find information about conjugacy classes of $HS$ stored in its character table in the following way:
gap> t:=CharacterTable("HS");
CharacterTable( "HS" )
gap> SizesConjugacyClasses(t);
[ 1, 5775, 15400, 123200, 11550, 173250, 693000, 88704, 147840, 1774080,
1232000, 1848000, 6336000, 2772000, 2772000, 2772000, 2217600, 2217600,
4032000, 4032000, 3696000, 2956800, 2217600, 2217600 ]
gap> OrdersClassRepresentatives(t);
[ 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8, 8, 10, 10, 11, 11, 12, 15,
20, 20 ]
gap> ClassNames(t);
[ "1a", "2a", "2b", "3a", "4a", "4b", "4c", "5a", "5b", "5c", "6a", "6b",
"7a", "8a", "8b", "8c", "10a", "10b", "11a", "11b", "12a", "15a", "20a",
"20b" ]
ConjugacyClasses
in the first instance. $\endgroup$