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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 $\\operatorname{M}(c,q^b)$. 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Corr, Cheryl E. Praeger","submitted_at":"2014-01-08T08:15:45Z","abstract_excerpt":"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 $\\operatorname{M}(c,q^b)$. 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