For a nilpotent Lie algebra L of dimension n and dim(where M (L) denotes the Schur multiplier of L. In case m = 1 the equality holds if and only if L ∼ = H(1) ⊕ A, where A is an abelian Lie algebra of dimension n − 3 and H(1) is the Heisenberg algebra of dimension 3.

We introduce the exterior degree of a finite group G to be the probability for two elements g and g in G such that g ∧ g = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier.

In virtue of a recent bound obtained in [P. Niroomand and F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011), 1293--1297], we classify all capable nilpotent Lie algebras of finite dimension possessing a derived subalgebra of dimension one. Indirectly, we find also a criterion for detecting noncapable Lie algebras. The final part contains a construction, which shows that there exist capable Lie algebras of arbitrary big corank (in the sense of Berkovich--Zhou)

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