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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 Möbius function of the subgroup lattice of a finite group G has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group G and denote by CA(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H]A = { Ha | a ∈ A} be the element of CA(G) containing H. We may define an ordering in CA(G) in the following way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A. We consider the Möbius function µA of the corresponding poset and analyse its properties and possible applications.

The key-scheduling algorithm in the AES is the component responsible for selecting from the master key the sequence of round keys to be xor-ed to the partially encrypted state at each iteration. We consider here the group F generated by the action of the AES-128 key-scheduling operation, and we prove that the smallest group containing F and all the translations of the message space is primitive. As a consequence, we obtain that no linear partition of the message space can be invariant under its action.

In this article, we characterize the genera of those quotient curves H q /G of the F q 2-maximal Hermitian curve H q for which either G is contained in the maximal subgroup M 1 of (H q ) fixing a self-polar triangle, or q is even and G is contained in the maximal subgroup M 2 of (H q ) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated to H q (F q 2) In this way, several new values for the genus of a maximal curve over a finite field are obtained. Our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [4] when G fixes a point P∈H q (F q 2) and q is even, and the open cases in [33] when G≤M 2 and q is odd

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

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group.

We define a translation based cipher over arbitrary finite field and we study properties of the permutation group generated by the round functions of such cipher. We show under which cryptographic hypothesis this group is primitive. Moreover, a minor strengthening of our hypothesis lets us prove that such group is the symmetric group or the alternating group for characteristic p bigger or equal to 2; in particular we improve a previous result

A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 whose automorphism group contains the symmetric group of degree 3, the alternating group of degree 4 or the dihedral group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5 or is isomorphic to the elementary abelian group of order 8

For a set X an unordered relation on X is a family R of subsets of X. If R is such a relation we let G(R) be the group of all permutations on X 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 X. 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.

For a set X an unordered relation on X is a family R of subsets of X. If R is such a relation we let G(R) be the group of all permutations on X 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 X. 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.

The group generated by the round functions of a block cipher has been widely investigated. We identify a large class of block ciphers for which this group is easily guaranteed to be primitive. Our class includes the AES cipher and the SERPENT cipher. © 2009 Springer-Verlag.

Let X be a set. Then an unordered relation on X is a family R of subsets of X. For such a family let G(R) be the group of all permutations g of X for which x^{g} belongs to R whenever x belongs to R. We are interested in permutation groups which can be represented in the shape G=G(R) for an unordered relation R on X

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. © 2009 Springer Science+Business Media, LLC.

We investigate properties of finite transitive permutation groups ( G, Omega) in which all proper subgroups of G act intransitively on Omega. In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups.

We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic, then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be torsion free if the dimension is infinite. We also give an example of an abelian, regular subgroup of the affine group over an infinite vector space, which intersects trivially the group of translations.

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.

We determine the minimally irreducible groups with proper non-soluble socle, whose degree is the product of two primes.

We compare the probability of generating with a given number of random elements two almost simple groups with the same socle S. In particular we analyse the case S = PSL(2,p).

We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.

A complete map for a group G is a permutation phi: G --> G such that g bar right arrow g phi (g) is still a permutation of G. A conjecture of M. Hall and L. J. Paige states that every finite group whose Sylow 2-subgroup is non-trivial and non-cyclic admits a complete map. In the present paper it is proved that a potential counterexample G of minimal order to this conjecture either is almost simple or G has only one involution, the Sylow 2-subgroups of G are quaternionic, \G/G\ less than or equal to 2, G' congruent to SL(2, q) for some odd prime power q > 5 and if G is not a perfect group then G/Z(G') congruent to PGL(2, q).

We determine the irreducible subgroups G of GL(r,C), where r is prime and all proper subgroups of G are reducible.

A structure theorem is proved for finite groups with the property that, for some integer m with m ≥ 2, every proper quotient group can be generated by m elements but the group itself cannot.