Absolutely faithful group actions

1969 ◽  
Vol 66 (2) ◽  
pp. 231-237 ◽  
Author(s):  
John S. Rose
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Wreath Product ◽  
Cyclic Group ◽  
Present Note ◽  
Group Actions ◽  
Conjugacy Classes ◽  
General Context

The definition and main result. It has been shown ((1), § 3) that if G is any finite group and p any prime number not dividing |G|, then the number of conjugacy classes of maximal nilpotent subgroups in the regular wreath product of a cyclic group of order p by G is equal to the number of conjugacy classes of all nilpotent subgroups in G. This fact, together with various properties of the map by means of which it was established, proved helpful in dealing with questions of construction raised in (1). The present note isolates the key property of the wreath product on which the argument rests, and from this shows how the argument can be carried over to a more general context. The essential situation is that a group G acts on a group A in a way which will be called ‘absolutely faithful’.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

scholarly journals On Finite Groups With ‘Hidden’ Primes

1971 ◽  
Vol 12 (3) ◽  
pp. 287-300 ◽  
Author(s):  
L. G. Kovács ◽  
Joachim Neubüser ◽  
B. H. Neumann
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Finite Groups ◽  
Infinite Order ◽  
Proper Subset ◽  
Minimal Set ◽  
Minimal Sets ◽  
Starting Point

The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G, must there be a minimal set of generators of G that contains an element whose order is divisible by p? A set of generators of G is minimal if no set with fewer elements generates G. A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powell's question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

scholarly journals A characterization of PGL (2,pn) by some irreducible complex character degrees

2016 ◽  
Vol 99 (113) ◽  
pp. 257-264 ◽  
Author(s):  
Somayeh Heydari ◽  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Group Algebra ◽  
Character Degrees ◽  
Complex Character ◽  
Complex Group ◽  
Complex Group Algebra ◽  
A Finite Group

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2,p) |, p ? cd(G) and max(cd(G)) = p+1, then G ? PGL(2,p), SL(2, p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also, we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G ? PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely determined by the structure of its complex group algebra.

scholarly journals Q-conjugacy character table for the non-rigid group of 2,3-dimethylbutane

2009 ◽  
Vol 74 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Reza Darafsheh ◽  
Ali Moghani
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Direct Product ◽  
Cyclic Group ◽  
Conjugacy Classes ◽  
Character Table ◽  
Irreducible Characters ◽  
Dominant Classes ◽  
A Finite Group ◽  
Rigid Group

Maturated and unmaturated groups were introduced by the Japanese chemist Shinsaku Fujita, who used them in the markaracter table and the Q-conjugacy character table of a finite group. He then applied his results in this area of research to enumerate isomers of molecules. Using the non-rigid group theory, it was shown by the second author that the full non-rigid (f-NRG) group of 2,3- -dimethylbutane is isomorphic to the group (Z3?Z3?Z3?Z3):Z2 of order 162 with 54 conjugacy classes. Here (Z3?Z3?Z3?Z3):Z2 denotes the semi direct product of four copies of Z3 by Z2, where Zn is a cyclic group of order n. In this paper, it is shown with the GAP program that this group has 30 dominant classes (similarly, Q-conjugacy characters) and that 24 of them are unmatured (similarly, Q-conjugacy characters such that they are the sum of two irreducible characters). Then, the Q-conjugacy character table of the unmatured full non-rigid group 2,3-dimethylbutane is derived.

scholarly journals A submatrix of the character table

2000 ◽  
Vol 62 (3) ◽  
pp. 525-528
Author(s):  
Gabriel Navarro
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Conjugacy Classes ◽  
Character Table ◽  
Maximum Rank ◽  
A Finite Group

Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.

THE CONJUGACY CLASSES OF A SUBGROUP $S_{n}^{m} : C_{m}$ OF Smn, prime m

2008 ◽  
Vol 18 (04) ◽  
pp. 705-717
Author(s):  
KENNETH ZIMBA ◽  
MERIAM RABOSHAKGA
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Direct Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Character Table ◽  
Split Extension ◽  
Alternative Method ◽  
Positive Integers

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and [Formula: see text] be the direct product of m copies of the symmetric group Sn of degree n. Then [Formula: see text] is a subgroup of the symmetric group Smn of degree m × n. Let g∈Smn, of type [mn] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of Sn act. Then g permutes the elements of the copies of Sn in [Formula: see text] and generates a cyclic group Cm = 〈g〉 of order m. The wreath product of Sn with Cm is a split extension or semi-direct product of [Formula: see text] by Cm, denoted by [Formula: see text]. It is clear that [Formula: see text] is a subgroup of the symmetric group Smn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of [Formula: see text], where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group [Formula: see text], this method is useful in the construction of Fischer–Clifford matrices of the group [Formula: see text]. These Fischer–Clifford matrices are useful in the construction of the character table of [Formula: see text].

scholarly journals Parameterization of Irreducible Characters for p-Solvable Groups

2012 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Outer Automorphisms ◽  
A Finite Group

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

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