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Let d(G) be the minimum number of elements required to generate a group G. For a group Gof order pnwith a derived subgroup of order pkand d(G)=d,we know the order of the Schur multiplier of Gis bounded by p12(d-1)(n-k+2)+1.In this paper, we find the structure of all p-groups that obtains the mentioned bound. Moreover, we show that all of them are capable.

Given a nilpotent Lie algebra Lof dimension ≤6on an arbitrary field of characteristic ≠2, we show a direct method to detect whether Lis capable or not via computations on the size of its nonabelian exterior square L∧L. For dimensions higher than 6, we show a result of general nature, based on the evidences of the low dimensional case, but also on the evidences of large families of nilpotent Lie algebras, namely the generalized Heisenberg algebras. Indeed, we detect the capability of L∧Lvia the size of the Schur multiplier M(L/Z∧(L))of L/Z∧(L), where Z∧(L)denotes the exterior center of L.

A peculiar structure is present in a finite group G, when D(G)={d(H,G)|HisasubgroupofG}is small enough (here d(H, G) denotes the relative commutativity degree). Recent contributions show that Ghas elementary abelian quotients, when |D(G)|≤4. We introduce a similar problem for the relative exterior degree d∧(H,G)and for the relative tensor degree d⊗(H,G). Theorems of structure are shown when Ghas a small number of relative tensor (or exterior) degrees. Among other things, we give new estimations for the gap d∧(H,G)-d⊗(H,G)and for the arithmetic average (d∧(H,G)+d⊗(H,G))/2.

The aim of this work is to find some exact sequences on the c-nilpotent multiplier of a group G. We also give an upper bound for the c-nilpotent multiplier of finite p-groups and give the explicit structure of groups whose take the upper bound. Finally, we will get the exact structure of the c-nilpotent multiplier and determine c-capable groups in the class of extra-special and generalized extra-special p-groups. It lets us to have a vast improvement over the last results on this topic.

We prove a theorem of splitting for the nonabelian tensor product $$L \otimes N$$L⊗Nof a pair (L, N) of Lie algebras Land Nin terms of its diagonal ideal $$L \square N$$L□Nand of the nonabelian exterior product $$L \wedge N$$L∧N. A similar circumstance was described few years ago in the special case $$N=L$$N=L. The interest is due to the fact that the size of $$L \square N$$L□Ninfluences strongly the structure of $$L \otimes N$$L⊗N.

Let Lbe a finite-dimensional nilpotent Lie algebra and d(L/Z(L)) be the minimal number generators for L/Z(L). It is known dimL/Z(L)=d(L/Z(L))dimL2-t(L)for an integer t(L)≥0.In this paper, we classify all finite-dimensional nilpotent Lie algebras Lwhen t(L)∈{0,1,2}.Moreover, we find a construction, which shows that there are Lie algebras of arbitrary t(L).

Recently the first two authors have introduced a group invariant, called exterior degree, which is related to the number of elements xand yof a finite group Gsuch that xΛy= 1 in the exterior square GΛGof G. Research on this topic gives some relations between this concept, the Schur multiplier and the capability of a finite group. In the present paper, we will generalize the concept of exterior degree of groups and we will introduce the multiple exterior degree of finite groups. Among other results, we will obtain some relations between the multiple exterior degree, multiple commutativity degree and capability of finite groups.

Recently the first two authors have introduced a group invariant, called exterior degree, which is related to the number of elements x and y of a finite group G such that xΛy = 1 in the exterior square GΛG of G. Research on this topic gives some relations between this concept, the Schur multiplier and the capability of a finite group. In the present paper, we will generalize the concept of exterior degree of groups and we will introduce the multiple exterior degree of finite groups. Among other results, we will obtain some relations between the multiple exterior degree, multiple commutativity degree and capability of finite groups.

By a well-known result of Green (Proc R Soc A 237:574–581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191–196, 1999), there is an integer t, say corank(G), such that $${|\mathcal{M}(G)| = p^{\frac{1}{2}n(n-1)-t}}$$. In Niroomand (J Algebra 322:4479–4482, 2009), the author showed for a non-abelian group G, corank(G) ≥ logp(|G|)−2 and classified the structure of all non-abelian p-groups of corank logp(|G|)−2. In the present paper, we are interesting to characterize the structure of all p-groups of corank logp(|G|)−1.