On the symmetry of images of word maps in groups

Cocke, William; Ho, Meng-Che

doi:10.1080/00927872.2017.1327065

Cited by 8 publications

(6 citation statements)

References 6 publications

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“…We note Theorem A shows that any subset of G that is closed under endomorphisms of G can occurs as the image of a word map w in v(F n ), it does not provide a description of w. However, it is possible in some cases to explicitly find w. We will show the following theorem which relates to the authors' earlier work on the chirality of groups [CH18].…”

Section: Introductionmentioning

confidence: 88%

Word maps in finite simple groups

Cocke

2019

Arch. Math.

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Elements of the free group define interesting maps, known as word maps, on groups. It was previously observed by Lubotzky that every subset of a finite simple group that is closed under endomorphisms occurs as the image of some word map. We improve upon this result by showing that the word in question can be chosen to be in any v(Fn) provided that v is not a law on the finite simple group in question. In addition, we provide an example of a word w that witnesses the chirality of the Mathieu group M 11 . The paper concludes by demonstrating that not every subset of a group closed under endomorphisms occurs as the image of a word map.

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Section: Introductionmentioning

confidence: 88%

Word maps in finite simple groups

Cocke

2019

Arch. Math.

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“…According to [5], a pair (G, w), where G is a group and w is a word, is called chiral if G w = G −1 w . The group G is called chiral if (G, w) is chiral for some w. Otherwise G is achiral.…”

Section: Rationality and Chiralitymentioning

confidence: 99%

“…The group G is called chiral if (G, w) is chiral for some w. Otherwise G is achiral. In [5], the authors comment that the existence of chiral groups follows from a result of Lubotzky [16]. They then began the process of classifying all finite chiral groups.…”

Section: Rationality and Chiralitymentioning

confidence: 99%

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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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“…) is chiral for some w. Otherwise G is achiral. In [5] the authors comment that the existence of chiral groups follows from a result of Lubotzky [15]. They then began the process of classifying all finite chiral groups.…”

Section: Rationality and Chiralitymentioning

confidence: 99%

Word problems for finite nilpotent groups

Camina¹,

Iñiguez²,

Thillaisundaram³

2020

Preprint

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Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that N w (1) ≥ |G| k−1 , where N w (1) is the number of k-tuples (g 1 , . . . , g k ) ∈ G (k) such that w(g 1 , . . . , g k ) = 1. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that N w (g) ≥ |G| k−2 , where g is a w-value in G, for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that N w (g) ≥ |G|, where g is a w-value in G for finite groups G of nilpotency class 2. In addition, for p a prime, we show that finite p-groups G, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for wordsFinally, 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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On the symmetry of images of word maps in groups

Cited by 8 publications

References 6 publications

Word maps in finite simple groups

Word maps in finite simple groups

Word problems for finite nilpotent groups

Word problems for finite nilpotent groups

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