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 _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum