Factorizations in Z_n and the Orders of GL(2, Z_n) and SL(2, Z_n)

摘要

For m, n > 0 and a in Z_n, let _mF_n(a) denote the number of ways that a can be expressed as the product of m-factors in Z_n. We derive formulas for _mF_n(a) in terms of the Euler Phi-Function. In the case k=2, we compute |GL(2, Z_n )| and |SL(2, Z_n )| in terms of 2-factorizations in Z_n.