Hi,
If I understand your question correctly, you want information on how
many elements of a finite group have each possible order. Although I
don't think there are built-in functions for this, it is easy to write
code for these. For instance:
OrderStatisticsPaired := function(G)
local L,D;
L := List(Set(G),Order);
D := DivisorsInt(Order(G));
return(List(D,x->[x,Length(Filtered(L,y->x=y))]));
end;;
This function takes as input a group and gives a list of ordered
pairs, for each divisor of the order of the group, how many elements
of the group have that order. For instance:
OrderStatisticsPaired(SymmetricGroup(4))
gives the output:
[ [ 1, 1 ], [ 2, 9 ], [ 3, 8 ], [ 4, 6 ], [ 6, 0 ], [ 8, 0 ], [ 12, 0 ], [ 24,
0 ] ]
indicating that in the symmetric group on four letters, there is one
element of the group of order one, nine elements of order two, eight
elements of order three, six elements of order four, and no element of
any higher order.
This exact function may not suit your needs but it's likely that some
variant of it will.
This code is inefficient for large groups; for such groups, one can
modify the code to only go over conjugacy classes instead of elements.
To create such a function, you can either paste the function code in
front of the GAP command prompt or include this in a file and then use
GAP's "Read" command to read that file in.
Vipul
* Quoting Sara Radfar who at 2011-01-06 05:14:32+0000 (Thu) wrote
> Hi
>
> I can find the same order elements of a group but can't find the set
> of the number of the same order elements of a group.Also how we can
> introduce a sporadic simple group to GAP?.For example $CO$.
>
> Thanks
> Sara
>
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