Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite
field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear
group and $U_{n}$ denote the unitriangular group of $n\times n$ upper
triangular matrices with ones on the diagonal, over the finite field $%
\mathbb{F}_{p}$. In fact $U_{n}$ is a Sylow $p$-subgroup of $GL_{n}(\mathbb{F}_{p})$ of order $p^{\frac{n(n-1)}{2}}$. Given $n_{p^{2}}$ be the number of elementary abelian p-subgroups of rank $2$ in $U_{n}$. How can we deduce the number of elementary abelian p-subgroups of rank $2$ in the whole linear group $GL_{n}(\mathbb{F}_{p})$?.
Conversely, given $N_{p^{2}}$ be the number of elementary abelian p-subgroups of rank $2$ in $GL_{n}(\mathbb{F}_{p})$. Is there a criterion deduces the number of elementary abelian p-subgroups of rank $2$ in $U_{n}$?.
In other words, what is the relationship between $N_{p^{2}}$ and $n_{p^{2}}$?.
Any help would be appreciated so much. Thank you all.
What you can certainly do is to determine the number of elementary abelian p-subgroups of rank 2 in GL(n,p) by means of computation. -- For example using the following GAP function:
NumberOfElementaryAbelianpSubgroupsOfRank2InGLnp := function ( n, p )
local G, S, cclS, cclG;
G := Image(IsomorphismPermGroup(GL(n,p)));
S := SylowSubgroup(G,p);
cclS := Filtered(ConjugacyClassesSubgroups(S),
cl->Size(Representative(cl))=p^2
and not IsCyclic(Representative(cl)));
cclG := List(EquivalenceClasses(cclS,
function(cl1,cl2)
return IsConjugate(G,Representative(cl1),
Representative(cl2));
end),
Representative);
cclG := List(cclG,cl->Representative(cl)^G);
return Sum(List(cclG,Size));
end;
Do you have a reason to believe that there is a relationship between your values N_{p^2} and n_{p^2} which can be described in some 'nice' way?
I suggest you follow http://turnbull.mcs.st-and.ac.uk/~martyn/5824/5824handouts.pdf
https://math.berkeley.edu/~apaulin/AbstractAlgebra.pdf
What you can certainly do is to determine the number of elementary abelian p-subgroups of rank 2 in GL(n,p) by means of computation. -- For example using the following GAP function:
NumberOfElementaryAbelianpSubgroupsOfRank2InGLnp := function ( n, p )
local G, S, cclS, cclG;
G := Image(IsomorphismPermGroup(GL(n,p)));
S := SylowSubgroup(G,p);
cclS := Filtered(ConjugacyClassesSubgroups(S),
cl->Size(Representative(cl))=p^2
and not IsCyclic(Representative(cl)));
cclG := List(EquivalenceClasses(cclS,
function(cl1,cl2)
return IsConjugate(G,Representative(cl1),
Representative(cl2));
end),
Representative);
cclG := List(cclG,cl->Representative(cl)^G);
return Sum(List(cclG,Size));
end;
Do you have a reason to believe that there is a relationship between your values N_{p^2} and n_{p^2} which can be described in some 'nice' way?
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