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2 results

The Reidemeister spectrum of finite abelian groups
Part of
- Structure and classification of infinite or finite groups
- Abelian groups
Pieter Senden
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 4 / November 2023

Published online by Cambridge University Press:

06 September 2023, pp. 940-959
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For a finite abelian group A, the Reidemeister number of an endomorphism φ is the same as the number of fixed points of φ, and the Reidemeister spectrum of A is completely determined by the Reidemeister spectra of its Sylow p-subgroups. To compute the Reidemeister spectrum of a finite abelian p-group P, we introduce a new number associated to an automorphism ψ of P that captures the number of fixed points of ψ and its (additive) multiples, we provide upper and lower bounds for that number, and we prove that every power of p between those bounds occurs as such a number.

A STEP BEYOND KNESER'S THEOREM FOR ABELIAN FINITE GROUPS
JEAN-MARC DESHOUILLERS, GREGORY A. FREIMAN
Journal:

Proceedings of the London Mathematical Society / Volume 86 / Issue 1 / January 2003

Published online by Cambridge University Press:

28 January 2003, pp. 1-28

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January 2003
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A precise description of a subset $\mathcal{A}$ of $\mathbb{Z} / n \mathbb{Z}$ satisfying
$$ | \mathcal{A} + \mathcal{A} | \leq 2.04 | \mathcal{A} | $$
is given. Basically, there exists a subgroup $\mathcal{H}$ of $\mathbb{Z} / n \mathbb{Z}$ such that $\mathcal{A}$ is included in an arithmetic progression of $\ell$ cosets modulo $\mathcal{H}$ and
$$(\ell - 1) | \mathcal{H} | \leq | \mathcal{A} + \mathcal{A} | - | \mathcal{A} |.$$
2000 Mathematical Subject Classification: 11B50, 11B83, 20E34.

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2 results

The Reidemeister spectrum of finite abelian groups

A STEP BEYOND KNESER'S THEOREM FOR ABELIAN FINITE GROUPS