A transitive permutation group with no fixed point free elements of prime order is called elusive. A permutation group on a set Ω is said to be 2-closed if G is the largest subgroup of
which leaves ...invariant each of the G-orbits for the induced action on
There is a conjecture due to Marušič, Jordan, and Klin asserting that there is no elusive 2-closed permutation group. In this article, we give a proof of the conjecture for permutation groups of degrees
and
where p, q, and r are (not necessarily distinct) three primes.
A nonidentity element of a permutation group is said to be semiregular provided all of its cycles in its cycle decomposition are of the same length. It is known that semiregular elements exist in ...transitive 2-closed permutation groups of square-free degree and in some special cases when the degree is divisible by a square of a prime. In this paper it is shown that semiregular elements exist in transitive 2-closed permutation groups of the following degrees
16p, where
is a prime,
where
is a prime,
12pq, where
are primes,
and either
or
18pq, where
are primes and
where
are primes, and
or qr < s, and
4pqrs, where
are primes, pqr < s,
and
As a corollary, a 2-closed transitive permutation group of degree
and different from 72 and 96 contains semiregular elements.
A finite permutation group Formula: see text is called Formula: see text-closed if Formula: see text is the largest subgroup of Formula: see text which leaves invariant each of the Formula: see ...text-orbits for the induced action on Formula: see text. Introduced by Wielandt in 1969, the concept of Formula: see text-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total Formula: see text-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group Formula: see text is said to be totally Formula: see text-closed if Formula: see text is Formula: see text-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally Formula: see text-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly Formula: see text totally Formula: see text-closed finite nonabelian simple groups: the Janko groups Formula: see text, Formula: see text and Formula: see text, together with Formula: see text, Formula: see text and the Monster Formula: see text. Moreover, if a finite totally Formula: see text-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely Formula: see text examples. In the course of obtaining this classification, we develop a general framework for studying Formula: see text-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.
A finite permutation group Formula: see text is called Formula: see text-closed if Formula: see text is the largest subgroup of Formula: see text which leaves invariant each of the Formula: see ...text-orbits for the induced action on Formula: see text. Introduced by Wielandt in 1969, the concept of Formula: see text-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total Formula: see text-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group Formula: see text is said to be totally Formula: see text-closed if Formula: see text is Formula: see text-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally Formula: see text-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly Formula: see text totally Formula: see text-closed finite nonabelian simple groups: the Janko groups Formula: see text, Formula: see text and Formula: see text, together with Formula: see text, Formula: see text and the Monster Formula: see text. Moreover, if a finite totally Formula: see text-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely Formula: see text examples. In the course of obtaining this classification, we develop a general framework for studying Formula: see text-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.
Fixed-point-free permutations, also known as derangements, have been studied for centuries. In particular, depending on their applications, derangements of prime-power order and of prime ...order have always played a crucial role in a variety of different branches of mathematics: from number theory to algebraic graph theory. Substantial progress has been made on the study of derangements, many long-standing open problems have been solved, and many new research problems have arisen. The results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the classification of the finite simple groups, to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.
The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism: a non-trivial automorphism whose cycles all have the same length. In this paper, we ...investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.
We consider whether a transitive solvable group contains a semiregular element (a fixed point free element whose orbits are all of the same length). We first construct new families of groups without ...semiregular elements. We also show that if n is a positive integer such that gcd(n, ϕ(n)) = 1, then every solvable group of degree n contains a semiregular element, where ϕ is Euler's phi function. As a consequence, we show that for such n if every quasiprimitive group of composite degree m dividing n is either A
m
or S
m
, then every transitive group of degree n contains a semiregular element.
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