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17. On overgroups of regular abelian p-groupsEdward Dobson, 2009, original scientific article Abstract: Let ▫$G$▫ be a transitive group of odd prime-power degree whose Sylow ▫$p$▫-subgroup ▫$P$▫ is abelian od rank ▫$t$▫. Weshow that if ▫$p > 2^{t-1}$▫, then ▫$G$▫ has a normal subgroup that is a direct product of ▫$t$▫ permutation groups of smaller degree that are either cyclic or doubly-transitive simple groups. As a consequence, we determine the full automorphism group of a Cayley diagraph of an abelian group with rank two such that the Sylow ▫$p$▫-subgroup of the full automorphism group is abelian.
Keywords: group theory, graph theory, Cayley graph, abelian group, regular group, p-group
Published in RUP: 15.10.2013; Views: 3232; Downloads: 157
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19. Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power orderEdward Dobson, 2010, original scientific article Abstract: We show that almost every Cayley graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of ▫$G$▫ (that is, ▫$G_L \triangleleft {\rm Aut}(\Gamma))$▫.
Keywords: mathematics, graph theory, Cayley graph, abelian group, automorphism group, asymptotic, ▫$p$▫-group
Published in RUP: 15.10.2013; Views: 4629; Downloads: 137
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