[Paper Review] Semisymmetric elementary abelian covers of the Möbius-Kantor graph

Aleksander Malnič, Dragan Marušič+2 more|ArXiv.org|Oct 18, 2005

Finite Group Theory Research10 references4 citations

TL;DR

This paper classifies all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph (Generalized Petersen graph GP(8,3)), using linear representations of automorphisms over prime fields. It establishes that no such covers exist for p=2, and provides exact counts of (p−1)/4 and 1+(p−1)/4 for p≡5,9,13,17,21(mod24), and (p+1)/4 and 1+(p+1)/4 for p≡3,7,11,15,23,19(mod24), respectively, with explicit voltage rules for each case.

ABSTRACT

Let $\p_N \colon X o X$ be a regular covering projection of connected graphs with the group of covering transformations isomorphic to $N$. If $N$ is an elementary abelian $p$-group, then the projection $\p_N$ is called $p$-elementary abelian. The projection $\p_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of $\Aut X$ lifts along $\p_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection $\p_N$ is minimal semisymmetric if $p_N$ cannot be written as a composition $\p_N = \p \circ \p_M$ of two (nontrivial) regular covering projections, where $\p_M$ is semisymmetric. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see {\em J. Algebr. Combin.}, {\bf 20} (2004), 71--97). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, the Generalized Petersen graph $\GP(8,3)$, are constructed. No such covers exist for $p =2$. Otherwise, the number of such covering projections is equal to $(p-1)/4$ and $1+ (p-1)/4$ in cases $p \equiv 5,9,13,17,21 (\mod 24)$ and $p \equiv 1 (\mod 24)$, respectively, and to $(p+1)/4$ and $1+ (p+1)/4$ in cases $p \equiv 3,7,11,15,23 (\mod 24)$ and $p \equiv 19 (\mod 24)$, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.

Motivation & Objective

• To classify all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, which is the Generalized Petersen graph GP(8,3).
• To determine for which primes p such covers exist and how many exist, depending on p modulo 24.
• To explicitly construct the voltage rules generating each such covering projection using linear algebra over finite fields.
• To identify the largest group that lifts in each case, distinguishing between groups H and M.
• To clarify that the covering graphs are not necessarily semisymmetric themselves, though the projections are minimal and semisymmetric.

Proposed method

• The method uses a linear representation of automorphisms acting on the first homology group of the graph, reducing the problem to finding invariant subspaces of matrix groups over prime fields.
• The analysis is conducted over the finite field Z_p, with the group of covering transformations being an elementary abelian p-group.
• Voltage rules are assigned to edges using vectors in Z_p^k (k=2 or 3), depending on the prime p modulo 24.
• The classification relies on group actions of Z_2×Z_2 on solutions to quadratic equations (e.g., α²+β²=−1 mod p or ν²=−i), which generate orbits of size four.
• The lifting of automorphism groups is analyzed by determining whether the largest group H or M lifts, depending on the parameters.
• The construction is based on the theory of regular covering projections and the lifting of subgroups of Aut(X) along the covering map.

Experimental results

Research questions

• RQ1 For which primes p does there exist a minimal semisymmetric elementary abelian regular covering projection of the Möbius-Kantor graph GP(8,3)?
• RQ2 How many pairwise nonisomorphic such covering projections exist for each prime p, and how does this number depend on p modulo 24?
• RQ3 What are the explicit voltage rules that generate each such covering projection?
• RQ4 Which automorphism group (H or M) lifts in each case, and how does this depend on the prime p and the parameters in the voltage assignment?
• RQ5 Are the resulting covering graphs themselves semisymmetric, or only the covering projections?

Key findings

• No minimal semisymmetric elementary abelian covering projections exist for p=2.
• For primes p≡5,9,13,17,21(mod24), the number of such covering projections is (p−1)/4.
• For primes p≡1(mod24), the number of such covering projections is 1+(p−1)/4.
• For primes p≡3,7,11,15,23(mod24), the number of such covering projections is (p+1)/4.
• For primes p≡19(mod24), the number of such covering projections is 1+(p+1)/4.
• The voltage rules are explicitly displayed for each case, with assignments in Z_p^2 or Z_p^3 depending on p modulo 24, and the projections come in orbits of size four under Z_2×Z_2 actions.