Some Discussions Of Minimal Nontrivial Actions And Inner Nilpotent Groups

A group is called an inner nilpotent group(or a minimal nonnilpotent group,Schimdt group)if every subgroup is nilpotent except itself.As we all know,such groups occur frequently when we consider a minimal counterexample in finite group theory.However,until 2005,three famous group theorists obtained a complete description of inner nilpotent groups by using the structural theorem of so-called minimal non-PST groups.But the proof is very complicated and is not a straightforward proof.The aim of this paper is to give a simple characterization of the structure of inner nilpotent groups from a new aspect and to investigate new information.First,we establish a criterion about irreducible automorphisms,which will play a significant technical role in subsequent discussion.Theorem 1.Let V be an elementary abelian p-group,which is regarded as a vector space over the finite field F = GF(p)of p elements.Assume that |V| = pn and a ? Aut(V)is a p?-automorphism of V.Then ? is irreducible on V if and only if the characteri,stic polynomial of a on V is irreducible.In particular,if o(a)= qe,where q?p is a prime,then a is irreducible if and only if n = ordqe(p)(that is n is the least positive integer such that pn ?1(mod qe))·Next,we improve the classical Hall-Higman's simplifying theorem to obtain a necessary and sufficient condition,which allow us to get a characterization of minimal nontrivial actions.Theorem 2.Let p be a prime,A is a p'-group acting via automorphisms on a non-trivial p-group P.Then this action is minimal non-trivial if and only if the following conditions hold:(1)Cp(A)? ?(P),where ?(P)is the Frattini subgroup of P;(2)the induced action of A on P/?(P)is irreducible.In fact,we can replace ?(P)by P' in above theorem.Theorem 2'Let p be a prime,A is a p'-group acting via automorphisms on a non-trivial p-group P.Then this action is minimal non-trivial if and only if the folloowing conditions hold:(1)Cp(A)?P?;(2)the induced action of A on P/P? is irreducible.We can obtain a new description of the structure of inner nilpotent groups by using Theorem 1 and Theorem 2.Theorem 3.A finite group G is an inner nilpotent group if and only if the following three conditions hold:(1)G = P×Q,P ?Sylp(G),and Q €Sylql(G)is cyclic,where p? q are primes;(2)CQ(P)=??(Q)?Cp(Q)= P? ? ?(P),where ?(Q)is the Frattini group of Q;(3)d(P)= ordq(p),where d(P)is the numbe7r of minimum generators of P(e-quivalently,v |P/?(P)| = pd(p)and ordq(p)is the least positive integer r such that pr ? 1(mod q).We get a construction of inner nilpotent groups of“minimal orders",which can be regarded as the common quotient of all inner nilpotent groups of orders with thesame set of prime divisors.Theorem 4.Let p ?q be primes and d = ordq(p).Let P = Cp x … x Cp be the elementary abelian group of order pd and Q =Cq be the cyclic group of order q.Then Q is isomorphic to a subgroup of Aut(P)and the corresponding semi-direct product P x Q is an inner nilpotent group.Furhermore,if G is an arbitrary inner nilpotent group of order paqb,then Px Q is isomorphic to a quotient of G.This implies that PxQ is of minimal order among all inner nilpotent {p,q}-groups.