Transitive permutation groups of prime-squared degree

Dobson, Edward ; Witte, Dave

Journal of Algebraic Combinatorics, Tome 16 (2002) no. 1, pp. 43-69.

Résumé

Summary: We explicitly determine all of the transitive groups of degree $p ^{2}, p$ a prime, whose Sylow $p$-subgroup is not isomorphic to the wreath product $\mathbb Z _{ p} \wr \mathbb Z _{ p}$ mathbbZ_p $\wr $mathbbZ_p . Furthermore, we provide a general description of the transitive groups of degree $p ^{2}$ whose Sylow $p$-subgroup is isomorphic to $\mathbb Z _{ p} \wr \mathbb Z _{ p}$ mathbbZ_p $\wr $mathbbZ_p , and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order $p ^{2}$, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order $p ^{2}$, and find all nonnormal Cayley graphs of order $p ^{2}$.

Détail

Classification : specifically,, the, classification, is, complete, for, any, prime, p,, such, that, p, /, in, 11,, 23, and, p, =, (qd, -, 1)/(q, -, 1),, for, every, prime-power, q, and, natural, number, d
Keywords: permutation group, Cayley graph, $p$-group