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).
