# Groups with a ternary equivalence relation

Article

## 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).

## Keywords

Group with collinearity Ternary equivalence relation Kinematic fibration

## Mathematics Subject Classification

Primary 51M10 Secondary 51G05

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