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October 20, 2025Open Access

On A-Groups with the Same Index Set as a Nilpotent Group

Key Points

The key finding is that if an A-group contains specific conjugacy class sizes, it must be abelian.
If the conjugacy class sizes include both |G||_p for all primes p and |G||, then G is abelian.
The analysis is based on properties of A-groups, particularly focusing on Sylow subgroups and their structure.
This result responds positively to a question from Camina and Camina, affirming long-standing theories about group classifications.

Abstract

Let G be a finite group and N (G) be the set of conjugacy class sizes of G. For a prime p, let |G||ₚ be the highest p-power dividing some element of N (G). and define |G|| = Π₏ ⏟ (₆) |G||ₚ. G is said to be an A-group if all its Sylow subgroups are abelian. We prove that if G is an A-group such that N (G) contains |G||ₚ for every p π (G) as well as |G||, then G must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006.

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Cite This Study

Zhou et al. (Wed,) studied this question.

synapsesocial.com/papers/68f6379bb481a140a36cf704 https://doi.org/https://doi.org/10.48550/arxiv.2506.15250

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