Alternating and symmetric groups with Eulerian generating graph

2017/05/23 by Lucchini, Andrea, Marion, Claude
#05C07 #05C45 #20B10 #20B35 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1705.08202

Abstract

Given a finite group G, the generating graph Γ(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 Γ(G) when G is an alternating group or a symmetric group. In particular, we determine the vertices of Γ(G) having even degree and show that Γ(G) is Eulerian if and only if n and n-1 are not equal to a prime number congruent to 3 modulo 4.

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