Of planar Eulerian graphs and permutations

Moran, Gadi

doi:10.1090/S0002-9947-1985-0766222-1

Of planar Eulerian graphs and permutations
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Trans. Amer. Math. Soc. 287 (1985), 323-341 Request permission

Abstract:

Infinite planar Eulerian graphs are used to show that for $v > 0$ the covering number of the infinite simple group ${H_v} = S/{S^v}$ is two. Here $S$ denotes the group of all permutations of a set of cardinality ${\aleph _v},{S^v}$ denotes its subgroup consisting of the permutations moving less than ${\aleph _v}$ elements, and the covering number of a (simple) group $G$ is the smallest positive integer $n$ satisfying ${C^n} = G$ for every nonunit conjugacy class $C$ in $G$.

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