[GAP Forum] Counting all elements in a group of a given order

[Stephen Linton]

Dear GAP Forum,



On 1 Jul 2010, at 19:11, Benjamin Sambale wrote:

> Size(Filtered(G,x->Order(x)=n)); counts the elements of order n in the group G.
> 

You can simplify this a little to

Number(G, x-> Order(x) = n)

and this will work quite efficiently for smaller groups.

For larger (and nonabelian) groups it would be more sensible to use the  conjugacy structure of the group. Something like:

Sum(Filtered(ConjugacyClasses(G), c-> Order(Representative(c)) = n), Size)

works well. For example for G = M_22 and n =4 this is some 50 times faster than the direct approach.

Finally, for the largest examples it might be best to factorise n into prime powers and use the Sylow subgroups to find representatives of all the conjugacy classes of elements of the appropriate prime power orders. Then, aving enumerated these elements, the number of elements of order n is simply the product.

	Steve


> Sandeep Murthy schrieb:
>> Hello.
>> 
>> Is there a direct function in GAP for counting all elements
>> in a group of a given order?
>> 
>> Sandeep.
>> 
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