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The Engel graph of a finite group $G$ is a directed graph encoding the pairs of elements in $G$ satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of almost simple groups are strongly connected. In this paper, we give a refinement to this analysis.

The M\"obius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the M\"obius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup $G$ of the general linear group $\mathrm{GL}(n,q)$ acting on the $n$-dimensional vector space $V=\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the finite field with $q$ elements. We find a relation between this function and the Euler characteristic of two simplicial complexes $\Delta_1$ and $\Delta_2$, the former raising from the lattice of the subspaces of $V$, the latter from the subgroup lattice of $G$., Comment: 10 pages. Revised version; the first version contained some inaccuracies that have been corrected

S. S. Magliveras et al. have described symmetric and public key cryptosystems based on logarithmic signatures (also known as group bases) for finite permutation groups. In this paper we show that if $G$ is a nontrivial finite group which is not cyclic of order a prime, or the square of a prime, then the round (or encryption) functions of these systems, that are the permutations of $G$ induced by the exact-transversal logarithmic signatures (also known as transversal group bases), generate the full symmetric group on $G$. This answers a question of S. S. Magliveras, D.R. Stinson and Tran van Trung., Comment: This is an old paper, published in 2006

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated by a couple of examples that might be of independent interest., Comment: 19 pages - minor revisions

W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(\rho(G))$, where $\rho$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(\rho(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G, +)$, what are the commutative ring structures $(G, +, \cdot)$ such that all automorphism of $G$ as a group are also automorphisms of $G$ as a ring?, Comment: 18 pages. Accepted for publication, Journal of Algebra, March 2017

We define a translation based cipher over an arbitrary finite field, and study the permutation group generated by the round functions of such a cipher. We show that under certain cryptographic assumptions this group is primitive. Moreover, a minor strengthening of our assumptions allows us to prove that such a group is the symmetric or the alternating group; this improves upon a previous result for the case of characteristic two., Comment: Extensive revision, including fixing a mistake concerning the blocks of imprimitivity

For a set $\Omega$ an unordered relation on $\Omega$ is a family R of subsets of $\Omega.$ If R is such a relation we let G(R) be the group of all permutations on $\Omega$ that preserves R, that is g belongs to G(R) if and only if x in R implies x^{g}\in R. We are interested in permutation groups which can be represented as G=G(R) for a suitable unordered relation R on $\Omega.$ When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group different from the Alternating Group and of degree bigger or equal to 11 is a relation groups. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprmitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation. This property is closely connected to the orbit closure of permutation groups. Since relation groups are orbit closed the results here imply that many classes of imprimitive permutation groups are orbit closed.

In a previous paper, we had proved that the permutation group generated by the round functions of an AES-like cipher is primitive. Here we apply the O'Nan Scott classification of primitive groups to prove that this group is the alternating group., Comment: 10 pages: several improvements in exposition and content, including ruling out the symmetric group 9 pages; several changes in layout and exposition

The group generated by the round functions of a block ciphers is a widely investigated problem. We identify a large class of block ciphers for which such group is easily guaranteed to be primitive. Our class includes the AES and the SERPENT.

We answer a question of Paterson, showing that all block systems for the group generated by the round functions of a key-alternating block cipher are the translates of a linear subspace. Following up remarks of Paterson and Shamir, we exhibit a connection to truncated differential cryptanalysis. We also give a condition that guarantees that the group generated by the round functions of a key-alternating block cipher is primitive. This applies in particular to AES., Comment: 9 pages - corrected embarrassing typo in the title