Search results for "Group Theory"

Product of nilpotent subgroups

2000

We will say that a subgroup X of G satisfies property C in G if \({\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}}\) for all \({g}\in G\). We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if \(N\triangleleft G\) is nilpotent and X is a nilpotent subgroup of G then \(C_G(N\cap X)\leqq X\) implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.

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.

On partial CAP-subgroups of finite groups

2015

Abstract Given a chief factor H / K of a finite group G, we say that a subgroup A of G avoids H / K if H ∩ A = K ∩ A ; if H A = K A , then we say that A covers H / K . If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding p k . If every subgroup of order p k , where k ≥ 1 , and every subgroup of order 4 (when p k = 2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.

Finite Group Elements where No Irreducible Character Vanishes

1999

AbstractIn this paper, we consider elements x of a finite group G with the property that χ(x)≠0 for all irreducible characters χ of G. If G is solvable and x has odd order, we show that x must lie in the Fitting subgroup F(G).

Characters and Sylow 2-subgroups of maximal class revisited

2018

Abstract We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.

Number of Sylow subgroups in $p$-solvable groups

2003

If G is a finite group and p is a prime number, let vp(G) be the number of Sylow p-subgroups of G. If H is a subgroup of a p-solvable group G, we prove that v p (H) divides v p (G).

Weights, vertices and a correspondence of characters in groups of odd order

1993

A Loopless Generation of Bitstrings without p Consecutive Ones

2001

Let F n (p) be the set of all n-length bitstrings such that there are no p consecutive ls. F n (p) is counted with the pth order Fibonacci numbers and it may be regarded as the subsets of {1, 2,…, n} without p consecutive elements and bitstrings in F n (p) code a particular class of trees or compositions of an integer. In this paper we give a Gray code for F n (p) which can be implemented in a recursive generating algorithm, and finally in a loopless generating algorithm.

A note on finite groups generated by their subnormal subgroups

2001

AbstractFollowing the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the subgroups.In this paper we provide an answer whenever $U$ is quasinilpotent and $\mathfrak{F}$ is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.AMS 2000 Mathematics subject classification: Primary 20F17; 20D35

On James Hyde's example of non-orderable subgroup of $\mathrm{Homeo}(D,\partial D)$

2020

In [Ann. Math. 190 (2019), 657-661], James Hyde presented the first example of non-left-orderable, finitely generated subgroup of $\mathrm{Homeo}(D,\partial D)$, the group of homeomorphisms of the disk fixing the boundary. This implies that the group $\mathrm{Homeo}(D,\partial D)$ itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.