Primary Cyclic Matrices in Irreducible Matrix Subalgebras

Primary Cyclic matrices were used (but not named) by Holt and Rees in their version of Parker's 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 $\operatorname{M}(c,q^b)$ be an irreducible subalgebra of $\operatorname{M}(n,q)$, where $n=bc >c$. We prove a generalisation of the Kung-Stong Cycle Index, and use it to obtain a lower bound for the proportion of primary cyclic matrices in $\operatorname{M}(c,q^b)$. This extends work of Glasby and the second author on the case $b=1$.