Abstract: In this paper we characterize the finite groups with an irredundant covering containing some p-Sylow subgroups. In particular we analyze the symmetric and alternating groups, finding their p-elements having a p-subgroup as centralizer.

Abstract: The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut(O7(3)) and Aut(S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

Abstract: A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts We then focus on the case when the normal subgroup is abelian Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d)

Abstract: Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let L=Ln(2) or Un(2), where n≧17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that Γ(G)=Γ(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of Ln(2). Also we conclude that the simple group Un(2) is quasirecognizable by element orders.

Abstract: In 1975, Sherman introduced the probability of an automorphism of a finite group, which fixes an arbitrary element of the group. In this paper we introduce a new probability concept, namely the probability of an automorphism of a given finite group such that it fixes a subgroup element of the group. Among other results, we construct some upper and lower bounds for both probabilities.

Abstract: A subgroup H of a finite group G is said to be S-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. In this paper we investigate the structure of finite groups that have some S-quasinormally embedded subgroups of prime-power order, and new criteria for p-nilpotency are obtained.

Abstract: We study finite groups which possess a strongly p-embedded subgroup for some odd prime p. The main results of the paper will be applied in the ongoing project to classify the simple groups of local characteristic p.

Abstract: A subgroup H of a given group G is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on H can be extended to some congruence on the entire group G. An arbitrary group G 1 is an HF subgroup of the direct product G 1 × G 2, as well as of the free product G 1 * G 2 of groups G 1 and G 2. In this paper a necessary and sufficient condition is obtained for a factor Gi of Adian’s n-periodic product Π ∈ G i of an arbitrary family of groups {G i } i∈I to be an HF subgroup. We also prove that for each odd n ≥ 1003 any noncyclic subgroup of the free Burnside group B(m, n) contains an HF subgroup isomorphic to the group B(∞, n) of infinite rank. This strengthens the recent results of A.Yu. Ol’shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group B(m, n) is SQ-universal in the class of all groups of period n. Moreover, it turns out that any countable group of period n is embedded in some 2-generated group of period n, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group B(m, n) is distinguished as a direct factor in any n-periodic group in which it is contained as a normal subgroup.

Abstract: In this article, we study the outer automorphism group of a group G decomposed as a finite graph of groups with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed.

Abstract: Let ϕ be an automorphism of prime order p of the group G with C G (ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.

Abstract: Let G be a p-group of nilpotency class k with finite exponent exp(G) and let m = blogp kc. We show that exp(M (c) (G)) divides exp(G)p m(k 1) , for all c 1, where M (c) (G) denotes the c- nilpotent multiplier of G. This implies that exp(M(G)) divides exp(G), for all finite p-groups of class at most p 1. Moreover, we show that our result is an improvement of some previous bounds for the exponent of M (c) (G) given by M. R. Jones, G. Ellis and P. Moravec in some cases.

Abstract: Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus $g\geq 3$ closed orientable surface at a prime $p\geq 5$ is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\Q$. As an application we find that, for any prime $p\geq 5$ a central extension of the genus $g$ mapping class group surjects onto the finite groups $\mathbb G_p(\Z/q\Z)$, for all but finitely many primes $q$. This method provides infinitely many finite quotients of a given mapping class group outside the realm of symplectic groups.

Abstract: Let G be a finite group of nilpotency class 2 and w a group word. In this short paper we show that the probability that a random n-tuple of elements from G satisfies w is at least one over the order of G. This answers a special case of a conjecture of Alon Amit.

Abstract: We study finite groups whose each primary subgroup is either subnormal or abnormal with respect to classes of all nilpotent, all p-closed, and all p-nilpotent groups. In particular, we fully describe these groups.

Abstract: A fixed-point-free group G of automorphisms of an abelian group is shown to be locally finite if any two elements of G generate a finite subgroup.

Abstract: Let K be a field, char(K) 6= 2. Suppose G = G(K) is the group of K-points of a reductive algebraic K-group G. Let G1 ≤ G be the group of K-points of a reductive subgroup G1 ≤ G. We study the structure of the normalizer N = NG(G1). In particular, let G = SL(2n,K) for n > 1. For certain well known embeddings of G1 into G, where G1 = Sp(2n,K) or SO(2n,K), we show that N/G1 ∼= μk(K), the group of k-th roots of unity in K. Here, k = 2n if certain Condition (♦) holds, and k = n if not. Moreover, there is a precisely defined subgroup N of N such that N/N ∼= Z/2Z if Condition (♦) holds, and N = N if not. Furthermore, when n > 1, as the main observations of the paper we have the following: (i) N is a self-normalizing subgroup of G; (ii) N ∼= G1 >⊳μn(K), the semidirect product of G1 by μn(K). Besides we point out that analogous results will hold for a number of other pairs of groups (G,G1). We also show that for the pair (g, g1), of the corresponding K-Lie algebras, g1 is self-normalizing in g; which generalizes a well-known result in the zero characteristic.

Abstract: Let G = NwrC n be the wreath product of N by C n , where N is a finite nilpotent group and C n is a cyclic group of order n Then the normalizer property holds for G

Abstract: We give a classification of maximal subgroups of odd index in finite groups whose socle is isomorphic to one of the groups PSL n (q), PSU n (q), or PSp n (q) for n ≥ 13.

Abstract: A characterization of the classes of all π-nilpotent, all π-closed, and all π-decomposable finite groups is obtained by using generalized subnormal Sylow subgroups.

Abstract: Let G be any of the (binary) icosahedral, generalized octahedral (tetrahedral) groups or their quotients by the center. We calculate the automorphism group Aut(G).