Let denote the family of all graph-group pairs where is 4-valent, connected and -oriented (-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs for which every nontrivial normal subgroup of has at most two orbits on the vertices of , and at least one normal subgroup has two orbits. In particular we show that has a unique minimal normal subgroup and that for a simple group and . This provides a crucial step towards a general description of the long-studied family in terms of a normal quotient reduction. We also give several methods for constructing pairs of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup .
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Keywords: Edge-transitive graphs, automorphism groups, oriented graphs, graph quotients, vertex-transitive graphs, quasiprimitive permutation groups, Cayley graphs.
@article{ALCO_2021__4_3_409_0, author = {Poznanovi\'c, Nemanja and Praeger, Cheryl E.}, title = {Four-Valent {Oriented} {Graphs} of {Biquasiprimitive} {Type}}, journal = {Algebraic Combinatorics}, pages = {409--434}, publisher = {MathOA foundation}, volume = {4}, number = {3}, year = {2021}, doi = {10.5802/alco.161}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.161/} }
TY - JOUR AU - Poznanović, Nemanja AU - Praeger, Cheryl E. TI - Four-Valent Oriented Graphs of Biquasiprimitive Type JO - Algebraic Combinatorics PY - 2021 SP - 409 EP - 434 VL - 4 IS - 3 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.161/ DO - 10.5802/alco.161 LA - en ID - ALCO_2021__4_3_409_0 ER -
Poznanović, Nemanja; Praeger, Cheryl E. Four-Valent Oriented Graphs of Biquasiprimitive Type. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 409-434. doi : 10.5802/alco.161. http://www.numdam.org/articles/10.5802/alco.161/
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