A Study On The Problem Of P - Group And The Characterization Of Limited Almost Single Group

Three open questions of p-groups and characterization of almost simple groups are studied in this thesis. The whole thesis is divided into five chapters.Chapter 1 Backgrounds and Main ResultsWe introduce background and main results. Some notations, concepts and def-initions are given too.Chapter 2 Three Open Questions of p-groupsIn section 1, we study locally nilpotent p-groups containing some kinds of special centralizers of subgroups. This topic is related to a question proposed by Blackburn in [2,§48, Problem 6] as following:To classify a non-abelian p-group G, such that there exists an element t of order p satisfying CG(t)=<t> ×C, where p>2 and C(?)Cpm, m>1.We study a locally nilpotent p-group G satisfying one of the following:1. There exists H≤G of order p2 or H an elementary abelian subgroup of order p3 such that Cg(H)=H;2. There exists x ∈G such that CG(x)=<x> xC, where |x|= p and C is cyclic of order pm, m>1.For the first case, we obtained:if G is locally nilpotent, and H is of order p2, we come to that G is an extension of a divisible abelian p-group of rank p-1 by a cyclic group of order p. If G is locally nilpotent and H is an elementary abelian subgroup of order p3, then G is an extension of a divisible abelian p-group of rank p-1 or 2p-2 by a finite p-group.For the second case, we obtained:if G is locally nilpotent, then G is an extension of a divisible abelian group of rank p-1 by a finite p-group.In the section 2, we study Question 237 proposed by Berkovich and Janko in [1]:To study a finite p-group G of which subgroups H generated by 2 elements satisfying |H|≤p2exp(H).We study finite p-group G of which subgroups H generated by 2 elements having maximal cyclic subgroup and obtain the nilpotent class of G, Exp(G’) and some properties of Gpi.In section 3, we study Question 805 proposed by Berkovich and Janko in [2]:To study p-group G such that each proper non-normal abelian subgroup A in G having its normal closure AG a minimal non-abelian subgroups.On this question, we study the case with stronger restriction and classify finite groups G with each proper non-normal subgroup A in G having its normal closure AG a minimal non-ableian subgroups.Chapter 3 OD-characterization of an Almost Simple GroupOD-characterization of a finite simple group is a topic proposed and started to study by A. R. Moghaddamfar in 2005. We continue to study this topic and prove that L7(3) is OD-characterized and GL7(3) is 3-OD-characterized. As a corollary, it is proved that PGL7(3) and SL7(3) are OD-characterized. It is worth to mention that GL7(3) has its prime graph connected, so the approach used in the discussion before is not effective.Chapter 4 Characterize Almost Simple Groups by Lengths of Con-jugacy ClassesIn 1987, J. G. Thompson proposed the following conjecture:Let G be a finite group with Z(G)=1, N(G)={n∈ N|G has a conjugacy class C such that|C|=n}. If M a non-commutative simple group such that N(G)= N(M), then G(?)M.This conjecture is only for simple groups and was studied by many mathemati-cians. In this Chapter, we discuss the case that M is not a simple group and prove that if M is the automorphism group of K3-group, then G(?)M.Chapter 5 Characterize Almost Simple Groups by Its Order and Largest Element OrderIn 1987, Wujie Shi proposed following conjecture:Let G be a finite group, M a non-abelian simple group, then G(?)M if and only if |G|=|S| and πe(G)=πe(S).Recent years, after the above conjecture was proved. It becomes an interesting topic to discuss if it is possible to finite simple group is determined by |S| and less number in πe(S). We investigate this topic too and successfully use the order of the group and largest element order to characterize K4-group of type L2(q) and K5-groups of type L3(p). We give an example to show that the automorphism groups of K4-groups of type L2(q) cannot be characterized by their orders and largest element orders.