scholarly journals Characterization of non-nilpotent groups with two irreducible character degrees

2005 ◽  
Vol 284 (1) ◽  
pp. 326-332 ◽  
Author(s):  
M. Bianchi ◽  
A. Gillio Berta Mauri ◽  
M. Herzog ◽  
Guohua Qian ◽  
Wujie Shi
Keyword(s):  
Irreducible Character ◽  
Nilpotent Groups ◽  
Character Degrees

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

scholarly journals Word problems for finite nilpotent groups

2020 ◽  
Vol 115 (6) ◽  
pp. 599-609
Author(s):  
Rachel D. Camina ◽  
Ainhoa Iñiguez ◽  
Anitha Thillaisundaram
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Character Degrees ◽  
Character Theory ◽  
Odd Order ◽  
Related Group

AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

Characterization of $${\textit{PGL}}(2 , p^{2} ) $$ by Order and Some Irreducible Character Degrees

2019 ◽  
Vol 46 (4) ◽  
pp. 1073-1081
Author(s):  
Ali Iranmanesh ◽  
Mozhgan Mokhtari ◽  
Abolfazl Tehranian
Keyword(s):  
Irreducible Character ◽  
Character Degrees

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

scholarly journals 𝑝-groups with exactly four codegrees

2020 ◽  
Vol 23 (6) ◽  
pp. 1111-1122
Author(s):  
Sarah Croome ◽  
Mark L. Lewis
Keyword(s):  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
Additional Hypothesis ◽  
Nilpotence Class ◽  
Irreducible Character Degree

AbstractLet G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(1)}. Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree {p^{2}}, {\lvert G:G^{\prime}\rvert=p^{2}}, or G has coclass at most 3, then G has nilpotence class at most 4. In the case of coclass at most 3, the order of G is bounded by {p^{7}}. With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6. In this case, the order of G is bounded by {p^{10}}.

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

scholarly journals A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989 ◽  
Vol 41 (1) ◽  
pp. 68-82 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Group Actions ◽  
Character Degrees ◽  
Prime Divisors ◽  
Irreducible Characters ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

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