In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the...

The injective chromatic number χi(G) [5] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. In this paper we define injective chromatic sum and injective strength of a graph and obtain the injective chromatic sum of complete graph, paths, cycles, wheel graph and complete bipartite graph. We a...

Derivations are discussed in Appendix B. The proofs of Theorems 1.1 and 1.2 both use tensor products. For those two proofs, the reader should be comfortable with the fact that injectivity and surjectivity of a linear map of vector spaces can be detected after a base extension: a linear map is injective or surjective if and only if its base extension to a larger field is injective or surjective....