Abstract
A group G has a Frobenius graphical representation (GFR) if there is a simple graph Γ whose full automorphism group is isomorphic to G acting on the vertices as a Frobenius group. In particular, any group G with a GFR is a Frobenius group and Γ is a Cayley graph. By very recent results of Spiga, there exists a function f such that if G is a finite Frobenius group with complement H and | G | > f (| H |) then G admits a GFR. This paper provides an infinite family of graphs that admit GFRs despite not meeting Spiga’s bound. In our construction, the group G is the Higman group A ( f , q 0 ) for an infinite sequence of f and q 0 , having a nonabelian kernel and a complement of odd order.
Highlights
Graphs and their automorphism groups have intensively been investigated especially for vertex-transitive graphs
Many contributions have concerned vertex-transitive graphs with large automorphism groups compared to the degree of the graph, and have in several cases relied upon deep results from Group theory, such as the classification of primitive permutation groups
A group is said to have a graphical regular representation (GRR) problem if there exists a graph whose automorphism group is isomorphic to G and acts regularly on the vertex-set
Summary
Graphs and their automorphism groups have intensively been investigated especially for vertex-transitive (and regular) graphs. A group is said to have a graphical regular representation (GRR) problem if there exists a graph whose (full) automorphism group is isomorphic to G and acts regularly on the vertex-set. Each graph Γ with a (sub)group G of automorphisms acting regularly on the vertex-set is a Cayley graph Cay(Γ(G, S)) All these give a motivation for the study of Frobenius groups G which have a graphical Frobenius representation (GFR), that is, there exists a graph whose (full) automorphism. Watskin in their recent paper [2] As it was pointed out by those authors, the GFR problem is largely not analogous to the GRR problem since all groups have a regular representation whereas Frobenius groups have highly restricted algebraic structures, and many large classes of abstract groups are not Frobenius groups.
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