Primary cyclic matrices in irreducible matrix subalgebras

Primary cyclic matrices were used (but not named) by Holt and Rees in their version of Parkers MEAT-AXE algorithm to test irreducibility of finite matrix groups and algebras. They are matrices X with at least one cyclic component in the primary decomposition of the underlying vector space as an X-module. Let M.c; qb/ be an irreducible subalgebra of M.n; q/, where n D bc > c. We prove a generalisation of the KungStong cycle index theorem, and use it to obtain a lower bound for the proportion of primary cyclic matrices in M.c; qb/. This extends work of Glasby and the second author on the case b D 1.