On the power graph of a certain gyrogroup

NEW

Problem: Reconstructed statement: for a gyrogroup $G$, with power graph $P(G)$ having vertex set $G$ and distinct $x,y$ adjacent iff $x=y^n$ or $y=x^n$ for some $n\ge1$, characterize when $P(G)$ is complete, especially when $G$ is non-degenerate, i.e. not a group. The source context is finite gyrogroups, but the wording omits finiteness; the result below covers both.

Result: No non-degenerate gyrogroup has a complete power graph.

More precisely, $P(G)$ is complete only if $G$ is actually a group. Hence, in the finite case, $P(G)$ is complete iff $G$ is a cyclic group of order $1$ or $p^m$. In the arbitrary infinite case, the additional complete-power-graph groups are the Prüfer $p$-groups $C_{p^\infty}$.

Proof. In every gyrogroup, the powers of a single element form an associative cyclic subgyrogroup: if $b=a^n$, then $\langle b\rangle\subseteq \langle a\rangle$.

Assume $P(G)$ is complete. For any $a,b\in G$, either $a=b$, or adjacency gives $b=a^n$ or $a=b^n$. Thus $\langle a\rangle$ and $\langle b\rangle$ are comparable by inclusion. Therefore any three elements $a,b,c$ lie in one cyclic subgyrogroup, namely a maximal member among $\langle a\rangle,\langle b\rangle,\langle c\rangle$. That cyclic subgyrogroup is a group, so

Thus $\circ$ is associative on all triples, so $G$ is a group. This contradicts non-degeneracy.

For completeness, the group classification is standard: a group with complete power graph has all cyclic subgroups linearly ordered. This forces all elements to have $p$-power order for one prime $p$. If element orders are bounded, the group is cyclic of order $p^m$; if unbounded, it is $C_{p^\infty}$. The converse is immediate from uniqueness of subgroups of each order in cyclic $p$-groups and Prüfer $p$-groups.

So the requested “condition” for a non-degenerate gyrogroup is impossible: there is no such $G$.

Citation: Uses the standard gyrogroup fact that cyclic subgyrogroups are groups; see Ungar, Analytic Hyperbolic Geometry, World Scientific, 2008, or Suksumran, “The Algebra of Gyrogroups,” 2016. The finite group criterion is Chakrabarty–Ghosh–Sen, Semigroup Forum 78 (2009), Theorem 2.12.