On the probability distribution associated to commutator word map in finite groups

Naik, Tushar Kanta

doi:10.1080/00927872.2019.1570239

Cited by 2 publications

(2 citation statements)

References 8 publications

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“…then there exist groups with |{N w k (g) : g ∈ G w k }| = n for all positive integers n. This is a recent result due to Naik [16].…”

Section: Introductionmentioning

confidence: 72%

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.

show abstract

“…then there exist groups with |{N w k (g) : g ∈ G w k }| = n for all positive integers n. This is a recent result due to Naik [16].…”

Section: Introductionmentioning

confidence: 72%

Word problems for finite nilpotent groups

Camina¹,

Iñiguez²,

Thillaisundaram³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…It is interesting to note that if instead of requiring |cd(G)| = 2 we require |cs(G)| = 2, then there exist groups with |{N w (g) : g ∈ G w }| = n for all positive integers n. This is a recent result due to Naik [17].…”

Section: Introductionmentioning

confidence: 95%

Word problems for finite nilpotent groups

Camina¹,

Iñiguez²,

Thillaisundaram

2020

Arch. Math.

View full text Add to dashboard Cite

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.

show abstract

On the probability distribution associated to commutator word map in finite groups

Cited by 2 publications

References 8 publications

Word problems for finite nilpotent groups

Word problems for finite nilpotent groups

Word problems for finite nilpotent groups

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