scholarly journals Walking around the Brauer tree

1974 ◽  
Vol 17 (2) ◽  
pp. 197-213 ◽  
Author(s):  
J. A. Green
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Defect Group ◽  
Finite Characteristic ◽  
Decomposition Matrix ◽  
Linear Graph ◽  
Famous Paper ◽  
A Finite Group

LetG be a finite group, and k a field of finite characteristic p, such that the polynomial x¦G¦ –1 splits completely in k[x]. Let Β be a kG-block which has defect group D which is cylclic of order pd (d ≧ 1). Brauer showed in a famous paper [2] that, in case d = 1, the decomposition matrix of Β is determined by a certain positive integer e which divides p − 1, and a tree Г, a connected acyclic linear graph of e + 1 vertices and e edges. Twenty-five years later Dade ([3]) extended Brauer's theorem to the general case.

scholarly journals Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

scholarly journals ON PERFECT ISOMETRIES FOR TAME BLOCKS

2002 ◽  
Vol 34 (1) ◽  
pp. 46-54 ◽  
Author(s):  
RADHA KESSAR ◽  
MARKUS LINCKELMANN
Keyword(s):  
Finite Group ◽  
Defect Group ◽  
Morita Equivalent ◽  
A Finite Group

Any 2-block of a finite group G with a quaternion defect group Q8 is Morita equivalent to the corresponding block of the centraliser H of the unique involution of Q8 in G; this answers positively an earlier question raised by M. Broué.

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

BOUNDING THE ORDER OF THE NILPOTENT RESIDUAL OF A FINITE GROUP

2016 ◽  
Vol 94 (2) ◽  
pp. 273-277
Author(s):  
AGENOR FREITAS DE ANDRADE ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Nilpotent Residual ◽  
A Finite Group

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

scholarly journals Almost Engel finite and profinite groups

2016 ◽  
Vol 26 (05) ◽  
pp. 973-983 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Profinite Group ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Nilpotent Residual ◽  
A Finite Group

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

The Structure of G/ Φ (G) in Terms of σ (G)

2006 ◽  
Vol 13 (01) ◽  
pp. 1-8
Author(s):  
Alireza Jamali ◽  
Hamid Mousavi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
A Finite Group

Let G be a finite group. We let [Formula: see text] and σ (G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/ Φ (G) when G is a finite soluble group with [Formula: see text].

scholarly journals Sums of element orders in groups of odd order

2021 ◽  
pp. 1-15
Author(s):  
Marcel Herzog ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cyclic Group ◽  
Element Orders ◽  
Maximum Value ◽  
Odd Order ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] denote the sum of element orders of [Formula: see text]. If [Formula: see text] is a positive integer let [Formula: see text] be the cyclic group of order [Formula: see text]. It is known that [Formula: see text] is the maximum value of [Formula: see text] on the set of groups of order [Formula: see text], and [Formula: see text] if and only if [Formula: see text] is cyclic of order [Formula: see text]. In this paper, we investigate the second largest value of [Formula: see text] on the set of groups of order [Formula: see text] and the structure of groups [Formula: see text] of order [Formula: see text] with this value of [Formula: see text] when [Formula: see text] is odd.

scholarly journals A Note on a Conjecture of Brauer

1963 ◽  
Vol 22 ◽  
pp. 1-13 ◽  
Author(s):  
Paul Fong
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Solvable Group ◽  
Prime Divisor ◽  
Irreducible Character ◽  
Defect Group ◽  
Irreducible Characters ◽  
A Finite Group

In [1] R. Brauer asked the following question: Let be a finite group, p a rational prime number, and B a p-block of with defect d and defect group . Is it true that is abelian if and only if every irreducible character in B has height 0 ? The present results on this problem are quite incomplete. If d-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that if is cyclic, then no characters of positive height appear in B. If is normal in , the conjecture was proved by W. Reynolds and M. Suzuki, [12]. In this paper we shall show that for a solvable group , the conjecture is true for the largest prime divisor p of the order of . Actually, one half of this has already been proved in [7]. There it was shown that if is a p-solvable group, where p is any prime, and if is abelian, then the condition on the irreducible characters in B is satisfied.

scholarly journals Minimal permutation representations of the two double covers of Sn

1996 ◽  
Vol 39 (2) ◽  
pp. 285-289
Author(s):  
John Brinkman
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Symmetric Group ◽  
Double Cover ◽  
Permutation Representation ◽  
Double Covers ◽  
A Finite Group

Let G be a finite group and denote by µ(G) (see [2]) the least positive integer m such that G has a faithful permutation representation in the symmetric group of degree m. This note considers the value of µ(G) when G is a double cover of the symmetric group.

scholarly journals ON THE EXPONENT OF A VERBAL SUBGROUP IN A FINITE GROUP

2012 ◽  
Vol 93 (3) ◽  
pp. 325-332 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Nilpotent Subgroup ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

AbstractLet $w$ be a multilinear commutator word. We prove that if $e$ is a positive integer and $G$ is a finite group in which any nilpotent subgroup generated by $w$-values has exponent dividing $e$, then the exponent of the corresponding verbal subgroup $w(G)$ is bounded in terms of $e$ and $w$only.

Sign in / Sign up

Export Citation Format

Share Document