Groups with a ternary equivalence relation

Abstract

We consider in a group $(G,\cdot )$ the ternary relation

$$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ | \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$

and show that $\kappa$ is a ternary equivalence relation if and only if the set $\mathfrak Z$ of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure

$$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ | \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$

We study the automorphism group of $(G,\kappa )$, i.e. all permutations $\varphi$ of the set G such that $(\alpha , \beta , \gamma ) \in \kappa$ implies $(\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa$. We show $\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)$, $\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )$ and if $\varphi \in \mathrm{Aut}(G,\kappa )$ with $\varphi (1)=1$ and $\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}$ for all $\xi \in G$ then $\varphi$ is an automorphism of $(G,\cdot )$. This allows us to prove a representation theorem of $\mathrm{Aut}(G,\kappa )$ (cf. Theorem 6) and that for $\alpha \in G$ the maps

$$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$

of the corresponding reflection structure $(G, \widetilde{G})$ (with $\tilde{G} := \{\tilde{\gamma }\ | \ \gamma \in G \}$) are point reflections. If $(G ,\cdot )$ is uniquely 2-divisible and if for $\alpha \in G$, $\alpha ^{1\over 2}$ denotes the unique solution of $\xi ^2=\alpha$ then with $\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}$, the pair $(G,\odot )$ is a K-loop (cf. Theorem 5).