Transitive permutation groups of prime-squared degree

Abstract

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 Zp ≀ Zp. Furthermore, we provide a general description of the transitive groups of degree p 2 whose Sylow p-subgroup is isomorphic to Zp ≀ Zp, 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 .

On Semiregular Elements of Solvable Groups

“…We remark that for the converse of the previous result, G can be chosen so that = 1 (see [11]). That is, we may choose G ≤ n/ .…”

Odd automorphisms in vertex-transitive graphs

“…Such is, for example, the case of vertex-transitive graphs of order a product of two primes, see [4,6,18,19,22], and the case of vertex-transitive graphs which are graph truncations, see [1]. The hard work needed to complete these classifications suggest that the even/odd question is by no means an easy one.…”

The isomorphism problem for Cayley ternary relational structures for some abelian groups of order 8p

“…Lemma 2.15 (Lemma 3, [8]). If there exists x ∈ G such that x(i, j) = (i + 1, αj + b i ), b i ∈ Z p , α ∈ Z * p , then {v(h) : h ∈ P 0 } is a cyclic code of length m over GF (p).…”