Journal of Algebra and Its Applications· 2026Q2
An infinite family of counterexamples to a question of Camina
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- Q2SCImago
- 2026year
Short summary
An infinite family of counterexamples to Camina's question on finite groups and their conjugacy class sizes has been constructed.
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Key points
- Constructs an infinite family of counterexamples to Camina's question.
- Addresses whether non-nilpotent groups with identical conjugacy class sizes to nilpotent groups must have a non-trivial center.
- Provides unconditional counterexamples, overcoming limitations of previous constructions dependent on Sophie Germain primes.
AI-generated from the title and abstract; the full text is not read.
Abstract
A.R. Camina and R.D. Camina posed the following question: Suppose there are two finite groups, one nilpotent and the other non-nilpotent, and the two groups share identical sets of conjugacy class sizes; must the non-nilpotent group possess a non-trivial center? Recently, W. Zhou gave a negative answer via a subtle and elegant construction of concrete counterexamples. Nevertheless, his approach relies on the existence of Sophie Germain primes, and thus fails to yield infinitely many counterexamples unconditionally. In the present paper, we construct an infinite family of counterexamples to Camina’s question.
The authors' abstract, as published at the source. Journal of Algebra and Its Applications, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics