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Both professions have a stake in the survival of statistics as a subject informed and structured by mathematics. To mathematics, statistics offers not only the example of an outward looking culture, but also entree to new problems ripe for mathematical study. To statistics, mathematics offers not only the safe harbor of organizational strength, but intellectual anchorage as well: mathematical understanding is an essential part of what distinguishes statistical thinking from most of the rest of information technology.

Enumeration avoids the unrealistic assumption of large expected cell counts, but unfortunately even computer-aided enumeration is practical only in small problems. There are simply too many possibilities. 2 X 10 18 tables have the same row and column totals. Alternatively, or in addition, there may be specific constraints that make enumeration difficult. For tables in three or more dimensions, for example, the hypothesis of interest may not be as simple as independence of the individual dimensions, and the constraints may involve several overlapping two-dimensional arrays of totals to be matched.

For every positive integer n, we have Zn' the group of integers modulo n. When n is prime, a simple application of Lagrange's theorem yields that this is the only group of order n. We may ask ourselves: what other positive integers have this property? In this spirit we call a positive integer n a cyclic number if every group of order n is cyclic. We define abelian and nilpotent numbers analogously. Recall that a group is nilpotent if and only if it is the (internal) direct product of its Sylow subgroups; see [7, 126].

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American Mathematical Monthly, volume 107, number 7, August-September 2000 by The Mathematical Association of America


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