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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]

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]

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]

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]

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]

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]

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]

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]

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 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: 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]