Search results for "Characteristic subgroup"
showing 10 items of 14 documents
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 ′ .
On the normal index of maximal subgroups in finite groups
1990
AbstractFor a maximal subgroup M of a finite group G, the normal index of M is the order of a chief factor H/K where H is minimal in the set of normal supplements of M in G. We use the primitive permutation representations of a finite group G and the normal index of its maximal subgroups to obtain results about the influence of the set of maximal subgroups in the structure of G.
Verbal sets and cyclic coverings
2010
Abstract We consider groups G such that the set of all values of a fixed word w in G is covered by a finite set of cyclic subgroups. Fernandez-Alcober and Shumyatsky studied such groups in the case when w is the word [ x 1 , x 2 ] , and proved that in this case the corresponding verbal subgroup G ′ is either cyclic or finite. Answering a question asked by them, we show that this is far from being the general rule. However, we prove a weaker form of their result in the case when w is either a lower commutator word or a non-commutator word, showing that in the given hypothesis the verbal subgroup w ( G ) must be finite-by-cyclic. Even this weaker conclusion is not universally valid: it fails …
C-Supplemented subgroups of finite groups
2000
A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that HKa G and H\ K is contained in CoreGOHU .W e follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.
Large subgroups of a finite group of even order
2011
It is shown that if G G is a group of even order with trivial center such that | G | > 2 | C G ( t ) | 3 |G|>2|C_{G}(t)|^{3} for some involution t ∈ G t\in G , then there exists a proper subgroup H H of G G such that | G | > | H | 2 |G|> |H|^{2} . If | G | > | C G ( t ) | 3 |G|>|C_{G}(t)|^{3} and k ( G ) k(G) is the class number of G G , then | G | ≤ k ( G ) 3 |G|\leq k(G)^{3} .
SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS
2007
We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG to contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs $P_2(6)$, …
The probability that $x$ and $y$ commute in a compact group
2010
We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…
On maximal subgroups of finite groups
1991
(1991). On maximal subgroups of finite groups. Communications in Algebra: Vol. 19, No. 8, pp. 2373-2394.
A characteristic subgroup and kernels of Brauer characters
2005
If G is finite group and P is a Sylow p-subgroup of G, we prove that there is a unique largest normal subgroup L of G such that L ⋂ P = L ⋂ NG (P). If G is p-solvable, then L is the intersection of the kernels of the irreducible Brauer characters of G of degree not divisible by p.