Search results for "Number"
showing 10 items of 3939 documents
Saturated formations and products of connected subgroups
2011
Abstract For a non-empty class of groups C , two subgroups A and B of a group G are said to be C -connected if 〈 a , b 〉 ∈ C for all a ∈ A and b ∈ B . Given two sets π and ρ of primes, S π S ρ denotes the class of all finite soluble groups that are extensions of a normal π-subgroup by a ρ-group. It is shown that in a finite group G = A B , with A and B soluble subgroups, then A and B are S π S ρ -connected if and only if O ρ ( B ) centralizes A O π ( G ) / O π ( G ) , O ρ ( A ) centralizes B O π ( G ) / O π ( G ) and G ∈ S π ∪ ρ . Moreover, if in this situation A and B are in S π S ρ , then G is in S π S ρ . This result is then extended to a large family of saturated formations F , the so-c…
On the Quadratic Type of Some Simple Self-Dual Modules over Fields of Characteristic Two
1997
Let G be a finite group and let K be an algebraically closed field of Ž characteristic 2. Let V be a non-trivial simple self-dual KG-module we . say that V is self-dual if it is isomorphic to its dual V * . It is a theorem of w x Fong 4, Lemma 1 that in this case there is a non-degenerate G-invariant alternating bilinear form, F, say, defined on V = V. We say that V is a KG-module of quadratic type if F is the polarization of a non-degenerate w x G-invariant quadratic form defined on V. In a previous paper 6 , the present authors described some methods to decide if such a module V is of w x quadratic type. One of the main results of 6 is the following. Suppose that Ž . G is a group with a s…
BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE
2013
We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.
Coding Partitions: Regularity, Maximality and Global Ambiguity
2007
The canonical coding partition of a set of words is the finest partition such that the words contained in at least two factorizations of a same sequence belong to a same class. In the case the set is not uniquely decipherable, it partitions the set into one unambiguous class and other parts that localize the ambiguities in the factorizations of finite sequences. We firstly prove that the canonical coding partition of a regular set contains a finite number of regular classes. We give an algorithm for computing this partition. We then investigate maximality conditions in a coding partition and we prove, in the regular case, the equivalence between two different notions of maximality. As an ap…
On extremal intersection numbers of a block design
1982
K.N. Majumdar has shown that for a 2-(v, k, @l) design D there are three numbers @a, @t, and @S such that each intersection number of D is not greater than @S and not less than max{@a, @t}. In this paper we investigate designs having one of these 'extremal' intersection numbers. Quasisymmetric designs with at least one extremal intersection number are characterized. Furthermore, we show that a smooth design D having the intersection number @S or @a>0 is isomorphic to the system of points and hyperplanes of a finite projective space. Using this theorem, we can characterize all smooth strongly resolvable designs.
POLYNOMIAL GROWTH OF THE*-CODIMENSIONS AND YOUNG DIAGRAMS
2001
Let A be an algebra with involution * over a field F of characteristic zero and Id(A, *) the ideal of the free algebra with involution of *-identities of A. By means of the representation theory of the hyperoctahedral group Z 2wrS n we give a characterization of Id(A, *) in case the sequence of its *-codimensions is polynomially bounded. We also exhibit an algebra G 2 with the following distinguished property: the sequence of *-codimensions of Id(G 2, *) is not polynomially bounded but the *-codimensions of any T-ideal U properly containing Id(G 2, *) are polynomially bounded.
Parabolic Subgroups of Artin Groups
1997
Abstract Let ( A , Σ) be an Artin system. For X ⊆ Σ, we denote by A X the subgroup of A generated by X . Such a group is called a parabolic subgroup of A . We reprove Van der Lek's theorem: “a parabolic subgroup of an Artin group is an Artin group.” We give an algorithm which decides whether two parabolic subgroups of an Artin group are conjugate. Let A be a finite type Artin group, and let A X be a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer of A X in A is the set of β in A such that β X β −1 = X . We prove that the commensurator of A X in A is equal to the normalizer of A X in A , and that this group is generated by A X and the quasi-centralizer of…
Centralizers of Parabolic Subgroups of Artin Groups of TypeAl,Bl, andDl
1997
Abstract Let ( A , Σ) be an Artin system of one of the types A l , B l , D l . For X ⊆ Σ, we denote by A X the subgroup of A generated by X . Such a group is called a parabolic subgroup of ( A , Σ). Let A X be a parabolic subgroup with connected associated Coxeter graph. We exhibit a generating set of the centralizer of A X in A . Moreover, we prove that there exists X ′ ⊆ Σ such that A X ′ is conjugate to A X and such that the centralizer of A X ′ in A is generated by the centers of all the parabolic subgroups containing A X ′ .
Injectors and Radicals in Products of Totally Permutable Groups
2003
Abstract Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.
Incomplete vertices in the prime graph on conjugacy class sizes of finite groups
2013
Abstract Given a finite group G, consider the prime graph built on the set of conjugacy class sizes of G. Denoting by π 0 the set of vertices of this graph that are not adjacent to at least one other vertex, we show that the Hall π 0 -subgroups of G (which do exist) are metabelian.