Generating functions for the powers in \(\mathrm{GL}(n, q)\)

Author name not available (Why is that?)

Abstract: Consider the set of all powers

e x t G L (n, q)^{M} = x^{M} m i d x i n e x t G L (n, q)

for an integer

M g e q 2

. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set

e x t G L (n, q)^{M}

, i.e., the elements or classes of these kinds which are

M^{t h}

powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when

(q, M) = 1

, (ii) for semisimple elements and all elements (and classes) when

M

is a prime power and

(q, M) = 1

, and (iii) for all kinds when

M

is a prime and

q

is a power of

M

.

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