[GAP Forum] A question about Omega

[Alexander Konovalov]

On 11 May 2011, at 15:07, 刘建军 wrote:

> Deat GAP Forum,
> 
> For a p-group $G$. The attibute "Omega(G,p)"(37.14) can be used when
> $G$ is abelian.
> Let $G$ be  a finite non-abelian p-group.
> Is there a method to get the subgroup that is generated by all elements of
> $G$ of order p?
> 
> Best Wishes
> Jianjun Liu

Dear Jianjun Liu, dear Forum,

There is no efficient general implementation but of course a naive 
approach may work dependently on the order and the representation of 
the group: compute conjugacy classes of elements, take a union of 
those of elements of order $p$ and generate a subgroup. Of course, 
it will have highly superfluous number of generators, but still you 
may be able to do operate with it and refine its generators to get 
rid of redundancies:

gap> G:=SmallGroup(512,213);
<pc group of size 512 with 9 generators>
gap> IsAbelian(G);
false
gap> cc:=Filtered(ConjugacyClasses(G),c->Order(Representative(c))=2);
[ f9^G, f8^G, f8*f9^G, f7^G, f7*f8^G, f4^G, f4*f8^G, f3*f5*f6^G, 
  f3*f5*f6*f8^G ]
gap> g:=Union(cc);
[ f4, f7, f8, f9, f4*f7, f4*f8, f4*f9, f7*f8, f7*f9, f8*f9, f3*f5*f6, 
  f4*f7*f8, f4*f7*f9, f4*f8*f9, f7*f8*f9, f3*f5*f6*f7, f3*f5*f6*f8, 
  f3*f5*f6*f9, f4*f7*f8*f9, f3*f5*f6*f7*f8, f3*f5*f6*f7*f9, 
  f3*f5*f6*f8*f9, f3*f5*f6*f7*f8*f9 ]
gap> H:=Subgroup(G,g);
<pc group with 23 generators>
gap> Size(H);
32
gap> IdGroup(H);
[ 32, 46 ]
gap> MinimalGeneratingSet(H);
[ f4, f7, f8, f3*f5*f6 ]
gap> StructureDescription(H);
"C2 x C2 x D8"

Hope this helps,
Alexander