For a finite group G, subgroup P of G is 2-minimal if B<P, where B=NG(S) for some Sylow 2-subgroup S and B contained in unique maximal P. Here we give detailed explicit description all the subgroups symplectic groups defined over field odd characteristic.

We introduce a strong form of Oliver’s p-group conjecture and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group Sn and of the general linear group GLn(Fq) satisfy both the strong conjecture and its reformulation.

— We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show fo...

A finite non-abelian group G is called metahamiltonian if every subgroup of either abelian or normal in G. If non-nilpotent, then the structure has been determined. nilpotent, determined by its Sylow subgroups. However, classification p-groups an unsolved problem. In this paper, are completely classified up to isomorphism.

Abstract We study the restriction to Sylow subgroups of irreducible characters symmetric groups. In particular, we give a precise description degrees constituents in terms shape partition that labels given character. Our main result is wide generalization [Giannelli and Navarro (Proc Am Math Soc 146(5):1963–1976, 2018), Theorem 3.1].

In this paper, we define a generalized Wielandt subgroup, local generalized Wielandt subgroup and its series for finite group and discuss its different basic properties which explain the notion of generalized Wielandt subgroup in a better way. We bound generalized Wielandt length as a function of nilpotency classes of its Sylow subgroups.

For a prime p > 2 and q = p, we compute a finite generating set for the SL2(Fq)-invariants of the second symmetric power representation, showing the invariants are a hypersurface and the field of fractions is a purely transcendental extension of the coefficient field. As an intermediate result, we show the invariants of the Sylow p-subgroups are also hypersurfaces.

Let G be a permutation group of set ? and k positive integer. The k-closure is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(?) which has same orbits as under componentwise action on ?k. It proved that finite nilpotent coincides with direct product k-closures all its Sylow subgroups.