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Valency
About: Valency is a research topic. Over the lifetime, 1632 publications have been published within this topic receiving 26141 citations.
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TL;DR: In this paper, a general method for computing the valency of any finite transitive group with stabiliser was presented and used to calculate the exact valency for a Frobenius group with cyclic kernel.
Abstract: Let $G$ be a permutation group on a set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted $\Sigma(G)$, with vertex set $\Omega$ and two vertices adjacent if they form a base. If $G$ is transitive, then $\Sigma(G)$ is vertex-transitive and it is natural to consider its valency (which we refer to as the valency of $G$). In this paper we present a general method for computing the valency of any finite transitive group and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.
TL;DR: In this paper, zero-field 55 Mn NMR spectra from two single crystals of La 1− x Sr 1+ x MnO 4 (x = 0 and 0.5) were presented.