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ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH
Published online by Cambridge University Press: 10 November 2017
Abstract
Given a finite group $G$, the generating graph
$\unicode[STIX]{x1D6E4}(G)$ of
$G$ has as vertices the (nontrivial) elements of
$G$ and two vertices are adjacent if and only if they are distinct and generate
$G$ as group elements. In this paper we investigate properties about the degrees of the vertices of
$\unicode[STIX]{x1D6E4}(G)$ when
$G$ is an alternating group or a symmetric group of degree
$n$. In particular, we determine the vertices of
$\unicode[STIX]{x1D6E4}(G)$ having even degree and show that
$\unicode[STIX]{x1D6E4}(G)$ is Eulerian if and only if
$n\geqslant 3$ and
$n$ and
$n-1$ are not equal to a prime number congruent to 3 modulo 4.
MSC classification
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2017
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