Let n be a positive integer, F a finite field of order q, and GL(n, F) the group of invertible n_n matrices with entries in F. We will be concerned with the order of the conjugacy classes in GL(n, F). Expressions for these values have been computed by Green (with the aid of an unpublished manuscript of Philip Hall) in [3]. In the following we will reconsider this problem by viewing it as an application of the Mo bius inversion theorem on an appropriate partially ordered set. In this direction, we compute the Mo bius function of a finite torsion module over a principal ideal domain, and note that this case includes the Mo bius functions of several often used partially ordered sets. Suppose T # GL(n, F ) and let [T ] denote its conjugacy class in GL(n, F).
is the index of the centralizer of T in GL(n, F). To obtain our desired result it is thus sufficient to compute the order of the centralizer of T in GL(n, F ). Our expression for this, which involves both binomial and q-binomial coefficients, appears rather different than the one obtained in Green. Related to the centralizer of T in GL(n, F ) is the classical result of Frobenius (see [5], e.g.) on the vector space dimension of the centralizer of T in M n (F), the ring of n_n F-matrices, and the results of Feit and Fine [1] on the number of pairs of commuting matrices in M n (F). Lemma 1 below can be thought of as a variation of Frobenius' result.
We begin our discussions by reviewing Mo bius inversion.
Let R be a commutative ring with identity and X a locally finite partially ordered set. Recall that a partially ordered set is locally finite if
and call this the interval between x and z.
If I(X, R)=[ f: X_X Γ R | f(x, y)=0 if x y] with the operations ( f + g)(x, y)= f (x, y)+ g(x, y) fg(x, y)= :
x z y f (x, z) g(z, y) (rf )(x, y)=rf (x, y)
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