The number of solutions of some equations in finite groups and a new proof of Itô's theorem

Tambour, Torbjörn

doi:10.1080/00927870008827160

Cited by 16 publications

(12 citation statements)

References 2 publications

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“…Thus, w ′ is equivalent under Aut (F d ) to a product of g disjoint commutators, where g is the genus of the surface it defines. Finding r amounts to finding this genus, as g = n we obtain a result from [Tam00]: for n odd we have W 1 = w 0 w n+1 w 2 w n+3 w 4 . .…”

Section: Corollaries and Applicationsmentioning

confidence: 99%

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On the Fourier expansion of word maps

Parzanchevski

Schul

2013

Bulletin of the London Mathematical Society

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Abstract. Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called its Fourier expansion. In this paper we present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones.

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Section: Corollaries and Applicationsmentioning

confidence: 99%

“…Let w be a word in , and the Fourier expansions of other words were studied as well (see [Isa94,chapter 4], and more recent results in [Tam00,AV11]). …”

Section: Introductionmentioning

confidence: 99%

On the Fourier expansion of word maps

Parzanchevski

Schul

2013

Bulletin of the London Mathematical Society

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“…where I is the identity matrix of size 1 (this is also proved by Tambour [10]). Multiplying this equality on the right by a −1 2 , and then summing over all a 2 a 3 a k ∈ G, we get…”

Section: The Main Resultsmentioning

confidence: 88%

“…Since the days of Frobenius, equations in finite groups have been studied by many authors who include Solomon [6], Isaacs [2], Kerber and Wagner [4], Strunkov [7,8], Tambour [10], etc. Strunkov [9] has summarized the connections between some questions in the representation theory of finite groups and in the theory of equations in groups.…”

Section: Introductionmentioning

confidence: 99%

On Solutions of a Class of Equations in a Finite Group

Das

Nath

2009

Communications in Algebra

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“…n ; see [19,24] respectively. We consider a version of Amit's conjecture as applied to general fibres:…”

Section: Introductionmentioning

confidence: 99%

Word problems for finite nilpotent groups

Camina¹,

Iñiguez²,

Thillaisundaram

2020

Arch. Math.

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Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that Nw(1) ≥ |G| k−1 , where for g ∈ G, the quantity Nw(g) is the number of k-tuples (g1,. .. , g k) ∈ G (k) such that w(g1,. .. , 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 Nw(g) ≥ |G| k−1 for g a w-value in G, and prove that Nw(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.

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The number of solutions of some equations in finite groups and a new proof of Itô's theorem

Cited by 16 publications

References 2 publications

On the Fourier expansion of word maps

On the Fourier expansion of word maps

On Solutions of a Class of Equations in a Finite Group

Word problems for finite nilpotent groups

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