Abstract: Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].

Abstract: We consider one-factorizations of K2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one-factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non-existence statements in case the number of fixed one-factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002

Abstract: In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

Abstract: The automorphism group of is studied, and a sufficient set of generators is given. Motivations for this theorem are given.

Abstract: We show that the automorphism group of the complex of pants decompositions for a surface is isomorphic to the mapping class group for that surface.

Abstract: The normalizer of the finite state automorphism group of a rooted homogeneous tree in the full automorphism group of this tree was investigated. General form of elements in the nor- malizer was obtained and countability of the normalizer was proved.

Abstract: Let X=Cay(G,S) be a 2-valent connected Cayley digraph of a regular p-group G and let G R be the right regular representation of G. It is proved that if G R is not normal in Aut(X) then X≅\(\)[2K 1 ] with n>1, Aut(X) ≅Z 2 wrZ 2n , and either G=Z 2n+1 = and S={a,a 2n+1 }, or G=Z 2n ×Z 2 = × and S={a,ab}.

Abstract: Suppose that we are given a set S of powers of a prime p and that 1 E S. A technique is presented that enables the construction of a p-group of specified nilpotence class n such that its set of irreducible character degrees is exactly S. If |S| ≥ 2, then this can be done for 2 ≤ n ≤ p and if p ∈ S, then the only requirement is 2 ≤ n.

Abstract: Let Fðr; cÞ be a free nilpotent group of rank r and class c, and let AutðFðr; cÞÞ be its automorphism group. A. Myasnikov [2, Problem N1] asked whether there are nontrivial elements of Fðr; cÞ which are fixed by all automorphisms of Fðr; cÞ. V. Bludov gave examples (see [2, Problem N1]) of such elements for r 1⁄4 2, c 1⁄4 4k, k 2. We classify (for r; c 2) all pairs ðr; cÞ for which Fðr; cÞ has nontrivial elements fixed by all automorphisrns (Theorem 5) and all pairs ðr; cÞ for which AutðFðr; cÞÞ has a nontrivial center (Theorem 6). Both theorems depend on a result of M. D. Burrow [3]. There is a natural action of SLðr;QÞ on the homogeneous components of the Lie subalgebra of the tensor algebra of Vr, where Vr 1⁄4 Q, and Burrow determined which homogeneous components contain nonzero SLðr;QÞ-invariants. We begin by recalling the classical results of I. Schur on the polynomial representations of GLðr;QÞ (for more details and unexplained terminology, see [4, pp. 37738] or [5, Chapter 5]). Let Vr 1⁄4 Q be the standard GLðr;QÞ-module, and let

Abstract: Let G be an abelian group and let K be a field of charK = p > 0. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the K-isomorphism KH ≅ KG for some group H yields H ≅ G provided G is a closed p-group or a p-local algebraically compact group. In particular, this is the case when G is closed p-primary of arbitrary power, or G is p-local algebraically compact with cardinality at most N 1 and K is in cardinality not exceeding N 1 . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

Abstract: We prove that if a finite group G of rank r admits an automorphism I� of prime order having exactly m fixed points, then G has a I�-invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1) Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3) The general case reduces modulo available results to the case of finite p-groups For reduction to Lie rings powerful p-groups are also used For them a useful fact is proved which allows us to "glue together" nilpotency classes of factors of certain normal series (Theorem 2)

Abstract: Let G be a finite group admitting an automorphism such that . If has prime order q then, by well-known results of Higman and Thompson, G is nilpotent of class bounded by some function depending on q alone. We extend this in the following way. Assume that is of square-free order n , prime to the order of G . Suppose that there exists a positive integer m such that for any non-trivial . Then G is nilpotent and the nilpotency class of G is bounded by some function depending only on m and n .

Abstract: The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.

Abstract: Sergei Nikolaevich Chernikov made an outstanding contribution to the investigation of groups with restrictions imposed on subgroups and, in particular, groups with various normality conditions for subgroups. In the present work, we continue investigations in this direction. We consider only finite groups. Let  be a nonempty, S-closed, saturated formation and let G be a group all subgroups of prime order of which are contained in the -hypercenter of Z G ( ). In [1], the structure of the group G was studied under the condition that it is solvable. In [2], a group G was not assumed to be solvable, but an additional condition was imposed on the formation . The aim of the present paper is to establish general properties of the group G that yield the results of [1, 2] as special cases. We also weaken the restriction imposed on , namely, instead of the saturation condition, we use the weaker condition of π-saturation. The notation used is standard [3, 4]. By π, we denote a fixed nonempty set of prime numbers, ′ π is the set of all prime numbers that do not belong to π, π G ( ) denotes the set of all prime divisors of the order of a group G, π π ( ) = ( ) ∈ G G U , and K (G) is the set of all simple groups isomorphic to the composition factors of a group G. If H/K is the normal section of G, then c H K G ( ) is the subgroup generated by all normal subgroups N of the group G such that N K ⊇ and K K N K H K ( ) ( ) I = ∅. Following Chunikhin, we say that a group whose order is divisible by numbers from π is a πd-group.

Abstract: The essay discusses the structure of group,which has the orders of pq ,and points out that p q groups have two sorts.

Abstract: In the paper, on the basis of the notion of saturation introduced in [1], we give a characterization of locally finite simple groups \(Re(P),Sz(Q),{\text{ and }}L_2 (P)\) with an Abelian Sylow 2-subgroup in the classes of mixed and periodic groups. A part of the main results was announced in [2]. Theorem 2 of the paper generalizes results of [3, 4].

Abstract: The transitivity of the automorphism group of a 2-(v,k,1) design heavily restricts the structure of the design. Buekenhout et al. determined almost all possible 2-(v,k,1) designs, of which the automorphism group act flag-transitively on them. After that people devote their attention to the designs with block-transitive automorphism groups. It is proved that if the automorphism group G of a 2-(v,k,1) design D acts block-transitively and point-primitively on D and the socle of G is an alternation group, then D is the projective geometry of dimension 3 over GF(2) and G≌A 7 or A 8.

Abstract: Given a regular action of a nite group G on a set V , we ask (and answer) the question of the existence of an incidence structure I = (V; B) on the set V whose full automorphism group Aut(I) is the group G in its regular action. Additional conditions on I also allow us to reene the original problem to the class of hypergraphs. Using results on graphical and digraphical regular representations ((4], 1]), we show the existence of a desired combinatorial structure (incidence structure or hypergraph) for all but a nite list of nite groups.

Abstract: Let be a prime integer and let be a free metabelian group of rank In the paper an automorphism of order permuting cyclically fixed bases of is studied. The subgroup of fixed points of and the properties of action of on the commutator subgroup of are described. There are also given some results concerning such automorphisms of free abelian groups of rank

Abstract: The classification of finite groups whose maximal subgroups are Dedekind groups is given with the following results:A finite group G whose maximal subgroups are Dedekind group are so if and only if G is (1) A Dedekind group;(2) The inner Abelian groups are of order p mq n ;(3) The inner 3 closed groups are of order 3 m2 3 ;(4) The inner Abelian p group except quaternion group;(5) The generalized quaternion groups are of order 2 4.

Abstract: The purpose of this paper is to classify the normal extensions G of a cyclic p-group , i.e., , with the irreducible character induced from a faithful linear character of . In fact we determine all normal extensions G of a cyclic p-group by p-subgroups of the automorphism group with the centralizer . Consequently, we have a theorem (Theorem 1) which might characterize the dihedral groups and the generalized quaternion groups of order among non-abelian 2-groups.

Abstract: This paper investigates the extent to which an Abelian group A is determined by the homomorphism groups Hom(A. G). A class C of Abelian groups is a Fuchs 34 class if A and C in C are isomorphic if and only if Hom(A,G)≅ Hom(C,G) for all G ∈ C. Two p-groups A and C satisfy Hom(A, G) ≅ Hoot(C, G) for all groups G if and only if they have the same n th -Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class G introduced by Glaz and Wickless. While G is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.

Abstract: . A group Gis called co-Dedekindianif every subgroup ofGis invar-iant under all central automorphisms of G. In this paper we give some necessaryconditionsforcertainfinitep-groupswithnon-cyclicabeliansecondcentretobeco-Dedekindian.Wealsoclassify3-generatorco-Dedekindianfinitep-groupswhichareofclass3,havingnon-cyclicabeliansecondcentrewithj 1 ðG p Þj ¼ p.2000MathematicsSubjectClassification.20E34,20D15,20D45.1. Introduction. Let Gbe a group, and let ZðGÞ denote the centre ofG.Anautomorphism of Gis called centralif x 1 ðxÞ2ZðGÞ for each x2 G. The set ofall central automorphisms of G, denoted by Aut c ðGÞ, is a normal subgroup of thefull automorphism group of G. A group Gis called co-Dedekindian(C-group forshort) if every subgroup of Gis invariant under all central automorphisms of G.In [1], Deaconescu and Silberberg give a Dedekind-like structure theorem for thenon-nilpotent C-groups with trivial Frattini subgroup and by reducing the finitenilpotentC-groupstothecaseofp-groupstheyobtainthefollowingtheorem.Theorem 1.1.LetGbeap-group.IfGisanon-abelianC-group,thenZ