Dear forum,
I have formulated and proved a certain proposition regarding collections
of nilpotent subgroups of a finite group. The expression "A < B" will stand
for
"A is a subgroup of B". If G is a finite group, then I will call F_G a
system of
nilpotent subgroups for G if F_G consists solely of nilpotent subgroups of
G,
and further satisfies:
1) F_G contains the trivial subgroup, and G is not in F_G;
2) if K is in F_G and J < K, then J is in F_G;
3) if K is in F_G, then K^g is in F_G for all g in G;
4) for each subgroup K of G (in particular, for G itself), the maximal
elements
of the sub-poset F_K := {S in F_G : S < K} (with inclusion as partial
order) forms
a single K-conjugacy class M_K;
5) for any two subgroups K, L of G with K < L, we have that (K:S) divides
(L:T),
where S is in M_K, and T is in M_L.
As an example, if G is not a p-group, and p divides |G|, for some fixed
prime p,
then we can take F_G to be the collection of all p-subgroups of G. Then
Sylow's
theorems guarantee that F_G is a system of nilpotent subgroups for G.
In fact, there is reason to speculate that if F_G is to be a system of
nilpotent
subgroups for a group G, then conditions 1)--5) are together strong enough
for
F_G to be some "well-known" family of subgroups (like Sylow), but I haven't
been
able to prove anything like that.
I would kindly ask for some help in formulating a GAP search for such
systems of
nilpotent subgroups.
Many thanks for taking the time to read my e-mail.
Best wishes,
Stefanos
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