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A p -group G is called a core- p 2 group if | H : H G | ≤ p 2 for every subgroup H of G (where H G denotes the normal core of H in G ). In this paper, it is proved that the nilpotent class of a finite core- p 2 p -group is at most 5 if p ≥ 3 , which is the best upper bound, and the derived length of a finite core- p 2 p -group is at most 3 if p ≥ 3 , which is also the best upper bound. [ABSTRACT FROM AUTHOR]

In this paper we investigate light dual multinets labeled by a finite group in the projective plane P G ( 2 , K ) defined over a field K . We present two classes of new examples. Moreover, under some conditions on the characteristic of K , we classify group-labeled light dual multinets with lines of length at least 9. [ABSTRACT FROM AUTHOR]

A finite group R is a DCI -group if, whenever S and T are subsets of R with the Cayley graphs Cay ( R , S ) and Cay ( R , T ) isomorphic, there exists an automorphism φ of R with S φ = T . Elementary abelian groups of order p 4 or smaller are known to be DCI -groups, while those of sufficiently large rank are known not to be DCI -groups. The only published proof that elementary abelian groups of order p 4 are DCI -groups uses Schur rings and does not work for p = 2 (which has been separately proven using computers). This paper provides a simpler proof that works for all primes. Some of the results in this paper also apply to elementary abelian groups of higher rank, so may be useful for completing our determination of which elementary abelian groups are DCI -groups. [ABSTRACT FROM AUTHOR]

We associate with every group G a set τ ( G ) of subgroups of G with 1 ∈ τ ( G ) . If H ∈ τ ( G ) , then we say that H is a τ-subgroup of G . If θ ( τ ( G ) ) ⊆ τ ( θ ( G ) ) for each epimorphism θ : G → G ⁎ , then we say that τ is a subgroup functor . We say also that a subgroup functor τ is: hereditary provided H ∈ τ ( E ) whenever H ≤ E ≤ G and H ∈ τ ( G ) ; regular provided for any group G , whenever H ∈ τ ( G ) is a p -group and N is a minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p ; Φ -regular (respectively Φ -quasiregular ) provided for any primitive group G , whenever H ∈ τ ( G ) is a p -group and N is a (respectively abelian) minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p . Let K ≤ H be subgroups of G and τ a subgroup functor. Then we say that: the pair ( K , H ) satisfies the F -supplement condition in G if G has a subgroup T such that H T = G and H ∩ T ⊆ K Z F ( T ) ; H is F τ -supplemented in G if for some τ -subgroup S ¯ of G ¯ contained in H ¯ the pair ( S ¯ , H ¯ ) satisfies the F -supplement condition in G ¯ , where G ¯ = G / H G and H ¯ = H / H G . In this paper we study the structure of a group G under the condition that some primary subgroups of G are F τ -supplemented in G . In particular, we prove the following result. Theorem A. Let F be a saturated formation containing the class U of all supersoluble groups, E a normal subgroup of G with G / E ∈ F , X = E or X = F ⁎ ( E ) , and τ a regular or hereditary Φ -regular subgroup functor. Suppose that every τ-subgroup of G contained in X is subnormally embedded in G. If every maximal subgroup of every non-cyclic Sylow subgroup of X is U τ -supplemented in G, then G ∈ F . Moreover, in the case when τ is regular, then every chief factor of G below E is cyclic. The results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results. [ABSTRACT FROM AUTHOR]

Let τ be a subgroup functor such that all subgroups of every finite group G contained in τ(G) are subnormal in G. In this paper, we give a simple proof of the fact that the lattice of all τ-closed totally composition formations of finite groups is inductive. [ABSTRACT FROM AUTHOR]

Abstract: In several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named -groups. Groups with all subgroups being -groups are called groups with the basis property. In the first part of this paper we correct and generalize some characterizations of groups with the basis property. In the second part we consider groups satisfying analogous conditions, but only for sets of generators of prime power orders. The class of groups introduced here is much larger then the class of groups with the basis property. For example, it contains all nilpotent groups. [Copyright &y& Elsevier]

Let G be a finite group. By MT(G) = (m1;, ... ;mk) we denote the type of conjugacy classes of maximal subgroups of G, which implies that G has exactly k conjugacy classes of maximal subgroups and m1;,...,;mk are the numbers of conjugates of maximal subgroups of G, where m1 ≤ ... ≤ mk. In this paper, we give some new characterizations of finite groups by the type of conjugacy classes of maximal subgroups. By πt(G) we denote the set of indices of all maximal subgroups of G. We also investigate the inuence of the set of indices of all maximal subgroups on the structure of finite groups. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring. In particular, we prove the following. Let k be a field, G a smooth k -group scheme of finite type, and X a quasi-compact quasi-separated locally Krull G -scheme. Assume that there is a k -scheme Z of finite type and a dominant k -morphism Z → X . Let φ : X → Y be a G -invariant morphism such that O Y → ( φ ⁎ O X ) G is an isomorphism. Then Y is locally Krull. If, moreover, Cl ( X ) is finitely generated, then Cl ( G , X ) and Cl ( Y ) are also finitely generated, where Cl ( G , X ) is the equivariant class group. In fact, Cl ( Y ) is a subquotient of Cl ( G , X ) . For actions of connected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected G . The proof depends on a similar result on (equivariant) Picard groups. [ABSTRACT FROM AUTHOR]

Suppose that G is an arbitrary Abelian group and A is any finite subset G. A set A is called a set with small sumset if, for some number K, we have | A + A| ≤ K| A|. The structural properties of such sets were studied in the papers of Freiman, Bilu, Ruzsa, Chang, Green, and Tao. In the present paper, we prove that, under certain constraints on K, for any set with small sumset, there exists a set Λ, Λ ≪ ɛ K log | A|, such that | A ν Λ| ≫ | A|/ K1/2+ ɛ, where ɛ > 0. In contrast to the results of the previous authors, our theorem is nontrivial even for a sufficiently large K. For example, for K we can take | A| η, where η > 0. The method of proof used by us is quite elementary. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the behaviour of uncountable groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to a given group class X . It is proved that if every proper subgroup of G of cardinality ℵ has finite conjugacy classes, then also the conjugacy classes of G are finite, provided that G has no simple homomorphic images of cardinality ℵ. Moreover, it turns out that if G is a locally graded group of cardinality ℵ in which every proper subgroup of cardinality ℵ contains a nilpotent subgroup of finite index, then G is nilpotent-by-finite, again under the assumption that G has no simple homomorphic images of cardinality ℵ. A similar result holds also for uncountable locally graded groups whose large proper subgroups are abelian-by-finite. [ABSTRACT FROM AUTHOR]

A subset D of a group G is a D-set if every element of G, not in D, has its inverse in D. Let A be a non-empty subset of G. A smallest D-set of G that contains A is called a D-set generated by A, denoted by 〈A〉. Note that 〈A〉 may not be unique. This paper characterized sets A with unique 〈A〉 and sets whose number of generated D-sets is equal to the index minimum. [ABSTRACT FROM AUTHOR]

In this paper we give a new characterization of simple group L2(p) with p a prime by both its order and nse(L2(p)), the set of numbers of elements of L2(p) with the same order. [ABSTRACT FROM AUTHOR]

Let π be a set of primes. By H. Wielandt's definition, Sylow π-theorem holds for a finite group G if all maximal π -subgroups of G are conjugate. In the paper, the following statement is proven. Assume that π is a union of disjoint subsets σ and τ and a finite group G possesses a π -Hall subgroup which is a direct product of a σ -subgroup and a τ -subgroup. Furthermore, assume that both the Sylow σ -theorem and τ -theorem hold for G . Then the Sylow π -theorem holds for G . This result confirms a conjecture posed by H. Wielandt in 1959. [ABSTRACT FROM AUTHOR]

Abstract: Let p be an odd prime and F be a saturated fusion system over a finite p-group S with derived subgroup of prime order, excepting the case when S ≅ P × A where P is a minimal nonabelian p-group with P ∩ Ω1(P) = 1, Ω1(P) is cyclic, and A is a finite abelian p-group. In this paper, we prove that S≤F. That is, S is resistant. This generalizes a result of Stancu in the odd prime case. [Copyright &y& Elsevier]

In this paper, we give the notions of crossed polymodule and cat¹-polygroup as a generalization of Loday's definition. Then, we define the pullback cat¹-polygroup and we obtain some results in this respect. Specially, we prove that by a pullback cat¹- polygroup we can obtain a cat¹ -group. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Fitting class and a formation. We call a subgroup of a finite group G the – -norm of G if is the intersection of the normalizers of the products of the -residuals of all subgroups of G and the -radical of G. Let π denote a set of primes and let denote the class of all finite π-groups. We call the subgroup of G the -norm of G. A normal subgroup N of G is called -hypercentral in G if either or and every G-chief factor below N of order divisible by at least one prime in π is -central in G. Let denote the -hypercentre of G, that is, the product of all -hypercentral normal subgroups of G. In this paper, we study the properties of the – -norm, especially of the -norm of a finite group G. In particular, we investigate the relationship between the -norm and the -hypercentre of G. [Copyright &y& Elsevier]

Let G be a finite group and πe(G) be the set of element orders of G. Let k ∈ πe (G)and mk be the number of elements of order k in G. Set nse(G):={mk|k ∈ πe (G)}. It is proved that An are uniquely determined by nse(An), where n ∈ {4,5,6}. In this paper, we prove that if G is a group such that nse(G)=nse(An) where n ∈ {7,8}, then G ≅ An. [ABSTRACT FROM AUTHOR]

Abstract: Cayley graphs are graphs constructed out of a finite group and its generating set . Circulant graphs are Cayley graphs corresponding to finite cyclic groups. In this paper, the value of the domination number for some circulant graphs is obtained and a corresponding dominating set is also determined. At last a necessary and sufficient condition for a subgroup to be an efficient dominating set in circulant graphs is also obtained. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and let be the set of all complex irreducible character degrees of G. Let be the set of all primes which divide some character degree of G. The prime graph attached to G is a graph whose vertex set is and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph has no triangles, then has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices. [Copyright &y& Elsevier]

Let G be a finite group and D(G) be the degree pattern of G. Denote by hOD(G) the number of isomorphism classes of finite groups H satisfying (|H|; D(H)) = (|G|; D(G)). A finite group G is called k-fold OD-characterizable if hOD(G) = k. As the main results of this paper, we prove that each of the following pairs {G1; G2} of groups: {Bn(q), Cn(q)}, n = 2M > 2, |π(qn + 1/2)| = 1, q is odd prime power; {Bp(3), Cp(3)}, |π(3p; - 1/2)| = 1, p is an old prime, {B3(5), C3(5)}, satisfies hOD(Gi) = 2, i = 1; 2. We also prove that, if (1) n = 2 and q is any prime power such that |π (q2 + 1/(2,q -1))| = 1 or (2) n = 2m ≥ 2 and q is a power of 2 such that |π(qn + 1, then hOD (Cn(q)) = hOD (Bn(q)) = 1. [ABSTRACT FROM AUTHOR]

Abstract: Two matrices , are called thin Lehman matrices if they are solutions of the matrix equation , where is the matrix of all s and is the identity matrix. These matrices are important in the set covering problem, but few examples are known. In this paper, we will introduce the notion of -overlapped factorizations of finite groups which constructs a new class of thin Lehman matrices. Moreover, we will study some structural properties of -overlapped factorizations. [Copyright &y& Elsevier]

Abstract: R. Baer and Wielandt in 1934 and 1958, respectively, considered the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this paper, for a finite group G, we define the subgroup to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set . Define for . By denote the terminal term of the ascending series. It is proved that if and only if the nilpotent residual is nilpotent. Furthermore, if all elements of prime order of G are in , then G is solvable and , where is p-length of G for , where means the set of prime divisors of . [Copyright &y& Elsevier]

The article discusses a paper that aims at using methods from homotopy theory to prove that some pairs of fusion systems of finite groups of Lie type are isotypically equivalent. It gives the formulations for the authors' main theorem resulting from the methods followed, comparing the fusion of groups of Lie type with different defining characteristics. Also included are a few proven cases of isotypical equivalences between fusion systems of classical groups.

Abstract: In this paper, we study the finite 2-group G such that for every . We prove that , is abelian and . We also get that . [Copyright &y& Elsevier]

Abstract: Let be finite abelian groups with nontrivial homomorphism group . Let be a non-empty subset of . Let denote the minimal integer, such that any sequence over of length must contain a nontrivial subsequence , such that for some . Let denote the minimal integer such that any sequence over of length must contain a nontrivial subsequence of length , such that for some . In this paper, we show that [Copyright &y& Elsevier]

Abstract: Let G be a finite p-solvable group and N be a normal subgroup of G. Suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. In this paper it is shown that, if H is a p-complement of N, then either H is abelian or H is a product of a q-group for some prime and a central subgroup of G. [Copyright &y& Elsevier]

Abstract: quiangular frames are an important class of finite dimensional frames because of their superior performance and numerous applications. The objective of this paper is to present a new tool to construct equiangular tight frames from groups. We prove that many equiangular tight frames arise from subsets of groups which we call signature sets. Subsequently, we define quasi-signature sets and examine real equiangular tight frames associated with these subsets of groups. This approach yields further results and establishes new correspondences. We extend these results to complex equiangular tight frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular tight frames that arise from groups in this manner. [ABSTRACT FROM AUTHOR]

Let G be a finite group and nse(G) the set of numbers of elements with the same order in G. In this paper, we prove that a finite group G is isomorphic to M, where M is one of the Mathieu groups, if and only if the following hold: (1) |G| = |M|, (2) nse(G) = nse(M). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that if and are subsets of a finite abelian group with , then , where and. In addition, we give a complete description of the subsets and of such that and . Our results generalize the corresponding theorems of Gallardo et al. in cyclic group [L. Gallardo, G. Grekos, L. Habsieger, et al., Restricted addition in and an application to the Erdös–Ginzburg–Ziv problem, J. London Math. Soc. 65 (2) (2002) 513–523]. [Copyright &y& Elsevier]

Abstract: Recently the author has shown that there are only finitely many p-groups with trivial Schur multiplicator of a given coclass if p is an odd prime. This paper introduces an algorithm to compute these finitely many p-groups for any given prime and coclass. As an application, we classify these groups for coclass 1, give a conjectural description for them for coclass 2 and determine them for coclass 3 and and coclass 4 and . Based on the experimental evidence, we conjecture that the order of a p-group with trivial Schur multiplicator and coclass r is bounded by a function which is independent of the prime p. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and an equivariant morphism of finite-dimensional G-modules. We say that φ is faithful if G acts faithfully on . The covariant dimension of G is the minimum of the dimension of taken over all faithful φ. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify groups of covariant dimension less than or equal to 2. A byproduct of our investigations is the existence of a purely transcendental field of definition of degree for a generic field extension of degree . [Copyright &y& Elsevier]

Let ω(G) denote the set of element orders of a finite group G. If H is a finite non-abelian simple group and ω(H) = ω(G) implies G contains a unique non-abelian composition factor isomorphic to H, then G is called quasirecognizable by the set of its element orders. In this paper we will prove that the group PSL4 (5) is quasirecognizable. [ABSTRACT FROM AUTHOR]

Abstract: Edmonds showed that two free orientation preserving smooth actions and of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class in the oriented bordism group of the group G. In this paper, we compute the bordism class for any such action of G on M and we determine for a given M, the bordism classes in that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form . [Copyright &y& Elsevier]

We consider finite groups only. Let ω be a nonempty subset of the set P of all primes, and let f : ω ∪ {ω′} → {formations of groups} and δ : P → {nonempty Fitting formations of groups} be some functions. The formation consisting of all groups G such that $${G \mathord{\left/ {\vphantom {G O}} \right. \kern-\nulldelimiterspace} O}_\omega (G) \in f(\omega ')$$ and $${G \mathord{\left/ {\vphantom {G G}} \right. \kern-\nulldelimiterspace} G}_{\delta (p)} \in f(p)$$ for any p ∈ ω ∩ π ( G) is referred to as an ω-fan formation with direction δ. Let ℌ be some class of groups; an ω-fan formation $$\mathfrak{F}$$ with direction δ is said to be an ℌωδ-critical formation if $$\mathfrak{F}\not \subseteq \mathfrak{H}$$ and any proper ω-fan subformation with direction δ in $$\mathfrak{F}$$ is contained in the class ℌ. In the paper, a description of the structure of the ℌωδ-critical formations is presented. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group and be the set of order components of G. Denote by the number of isomorphism classes of finite groups H satisfying . It is proved that some finite groups are uniquely determined by their order components, i.e. . Let . As the main result of this paper, we prove that if q is odd, then and if q is even, then . A main consequence of our results is the validity of a conjecture of J.G. Thompson and another conjecture of W. Shi and J. Bi for the groups , where and q is even. [Copyright &y& Elsevier]

Let G be a finite group and ω ( G ) be the set of element orders of G. Let k ∈ ω ( G ) and m k be the number of elements of order k in G. Let n s e ( G ) = { m k | k ∈ ω ( G ) } . In this paper, we prove that if G is a finite group such that nse(G) = nse(H), where H = P S U ( 3 , 3 ) or P S L ( 3 , 3 ) , then G ≅ H . [ABSTRACT FROM AUTHOR]