Let G be a group. A set X of elements of G is said to be non-commuting if xy 0 yx for any x; y A X with x 0 y. If jX j d jX 0 j for any other non-commuting set X 0 in G, then X is said to be a maximal non-commuting set. In this paper we obtain upper and lower bounds for the cardinality of a maximal non-commuting set in an extraspecial p-group where p is an odd prime.

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

Let R be an associative ring with identity. An element x ∈ R is said to be weakly clean if x = u + e or x = u − e for some unit u and idempotent e in R. The ring R is said to be weakly clean if all of its elements are weakly clean. In this paper we obtain an element-wise characterization of abelian weakly clean rings. A relation between unit regular rings and weakly clean rings is also obtained.

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