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 ^{1over 2}) denotes the unique solution of
(xi ^2=alpha ) then with
(alpha odot beta := alpha ^{1over 2} cdot beta cdot alpha ^{1over 2}), the pair
((G,odot )) is a K-loop (cf. Theorem 5).
