Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan–Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic 2. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent Lie algebras correspond to orbits of subspaces under this action. In some cases, these orbits are determined using geometric invariants, such as the Gram determinant or the Arf invariant. As a byproduct, we completely determine, for a 4-dimensional vector space V , the orbits of GL(V ) on the set of 2-dimensional subspaces of V ∧ V .

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of 'Cartesian decompositions' of the permuted set, relating them to certain 'Cartesian systems of subgroups'. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.

Let P be a partition of a finite set X. We say that a full transformation f : X −→ X preserves (or stabilizes) the partition P if for all P ∈ P there exists Q ∈ P such that P f ⊆ Q. Let T (X, P) denote the semigroup of all full transformations of X that preserve the partition P.In 2005 Huisheng found an upper bound for the minimum size of the generating sets of T (X, P), when P is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture.The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of T (X, P), when P is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then G, a \ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elements g ∈ G generate a semigroup denoted a g | g ∈ G . We classify the finite permutation groups G on a finite set X such that the semigroups G, a , G, a \G, and a g | g ∈ G are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups G, a \ G and a g | g ∈ G are generated by their idempotents for all non-invertible transformations of X.

The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.

We show that for every integer k ≥ 3 and every prime p ≤ 5 there is a group with soluble length k and order .

We define the entropy of a graded algebra A = n A n by lim sup n→∞ n dim A n . This is related to the notion of entropy in symbolic dynamics, and could serve as a natural dimension concept for algebras with exponential growth. We study the entropy of quotients and subalgebras of free associative algebras and free Lie algebras. We also study the behavior of the entropy function under free products, and obtain several characterizations of free algebras in terms of entropy.