The primary purpose of this paper is to investigate some commutator conditions for rings, which were suggested by group-theoretic results of F, W. Levi, I. D. Macdonald and N. D. Gupta. Most of these conditions can be simply interpreted in terms of inner derivations, and they suggest further questions about arbitrary derivations.In

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The nonabelian tensor square G[otimes ] G of a group G is generated by the symbols g[otimes ] h, g,h ∈ G, subject to the relations $$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime \in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4] and [5], extending ideas of J.H.C. Whitehead in [10]. The topic of this paper is the classification of 2-generator 2-groups of class two up to isomorphism and the determination of nonabelian tensor squares for these groups.

The following conditions for a group G are investigated:(i) maximal class n subgroups are normal, (ii) normal closures of elements have nilpotency class n at most, (iii) normal closures are n-Engel groups, (iv) G is an (n+l)-Engel group. Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1 . Here the question is settled for n = 2 as follows: conditions ( i i ) , (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.

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ABSTRACT. A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ(G). Determining σ(G) is an open problem for many non-solvable groups. For symmetric groups S n , Maróti determined σ(S n ) for odd n with the exception of n = 9 and gave estimates for n even. In this paper we determine σ(S n ) for n = 8, 9, 10 and 12. In addition we find the covering number for the Mathieu group M 12 and improve an estimate given by Holmes for the Janko group J 1 .

In this paper we determine the nonabelian tensor square of the free 2-Engel group of rank 3 and of the Burnside group on 3 generators of exponent 3. Both tensor squares are nilpotent groups of class 2. The calculatory method used is based on the concept of a crossed pairing. Some of the expansion formulas and verifications occuring in this context require extensive calculations. A computer program written in the GAP language assisted in completing these symbolic computations.