In this paper we continue developing the theory of symplectic\ud alternating algebras that was started in Traustason (2008). We focus on nilpotency, solubility and nil-algebras. We show in particular that symplectic alternating nil-2 algebras are always nilpotent and classify all nil-algebras of dimension up to 8

Let m, n be positive integers, v a multilinear commutator word and w = v m . We prove that if G is a residually finite group in which all wvalues are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent. then x is a (left) n-Engel element. A group G is called n-Engel if all elements of G are n-Engel. It is a long-standing problem whether any n-Engel group is locally nilpotent. Following Zelmanov's solution of the restricted Burnside problem [16,17], Wilson proved that this is true if G is residually finite [14]. Later the second author showed that if in a residually finite group G all commutators [x 1 , . . . , x k ] are n-Engel, then the subgroup [x 1 , . . . , x k ] | x i ∈ G is locally nilpotent [10,11]. This suggests the following conjecture.Conjecture. Let w be a group-word and n a positive integer. Assume that G is a residually finite group in which all w-values are n-Engel. Then the corresponding verbal subgroup w(G) is locally nilpotent. Preliminary resultsGiven subgroups X and Y of a group G, we denote by X Y the smallest subgroup of G containing X and normalized by Y .Lemma 2.1. Let x and y be elements of a group G satisfying [x, n y m ] = 1, for some n, m ≥ 1. Then x y is finitely generated.Proof. Set X = x y m . Then X is finitely generated by [9, Exercise 12.3.6]. Since x y = X y i | i = 0, . . . , m − 1 , the lemma follows.Corollary 2.2. Let y be an element of a group G and H a finitely generated subgroup. If y m is Engel for some m ≥ 1, then H y is finitely generated.The following lemma is well-known. We supply the proof for the reader's convenience.

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We provide two sufficient conditions to guarantee that the round functions of a translation based cipher generate a primitive group. Furthermore, under the same hypotheses, and assuming that a round of the cipher is strongly proper and consists of m-bit S-Boxes, with m = 3, 4 or 5, we prove that such a group is the alternating group. As an immediate consequence, we deduce that the round functions of some lightweight translation based ciphers, such as the PRESENT cipher, generate the alternating group.

We consider a finiteness condition on centralizers in a group G, namely that |C G (x)