Abstract :
A group image of complex square invertible matrices is said to have submultiplicative spectrum if for any G, H in image we have that σ(GH)subset ofσ(G)σ(H), where σ(G) denotes the spectrum of G. The question whether such a group can be irreducible arises in the study of the structure of matrix semigroups and is answered in the positive for groups with odd exponent (cf. Radjavi and Rosenthal, Simultaneous Triangularization, Springer, New York, 1999). Here we consider the most important remaining case, 2-groups with submultiplicative spectrum. If there is an element in the group of the kind whose determinant has order equal to either the exponent or half of the exponent of the group, then the group is reducible. We give an example of an irreducible group of the kind such that the order of determinant of every element equals one quarter of the exponent of the group.
