This is a somewhat vague question: for a prime number *p*, we often see that various counts come out to be 1 modulo *p*. What are the possible reasons for this?

Here are some I've encountered:

- For some reason, the counting can be reduced modulo
*p*, or in the field of*p*elements, and in that setting, the answer turns out to be 1. - There is a group action of a
*p*-group on the set with exactly one fixed point. Hence, all the other orbits have sizes equal to a power of*p*. This is used, for instance, in proving that the number of*p*-Sylow subgroups is congruent to*1*mod*p*. - A kind of inductive counting argument, that is based on the following observation: the sum of
*a*numbers, each of which is 1 mod*p*, is congruent to*a*mod*p*. Thus, the sum is 1 mod*p*iff*a*is*1*mod*p*. This comes up, for instance, in inductive proofs that the number of subgroups of order $p^k$ in a group of order $p^n$ is congruent to 1 mod $p$. A more general version of this is Phillip Hall's Enumeration Theorem. (This argument can be viewed as a variant of (1), but I find it sufficiently distinctive to mention separately). - The number comes up as the number of one-dimensional subspaces of a finite-dimensional vector space over a field with
*p*elements, which we know is a polynomial in*p*with constant term*1*. - More sophisticated combinations of (3) and (4) are used to prove that the number of subgroups of various kinds in
*p*-groups is congruent to 1 mod*p*. See, for instance, the work of Jonah and Konvisser:*Counting abelian subgroups of p-groups: a projective approach*,*Journal of Algebra*, Page 309-330, 1975. - An application of Fermat's little theorem or a related "order in the multiplicative group" result, wherein a prime
*q*distinct from*p*can divide $(a^p - 1)/(a - 1)$ only if it is congruent to 1 mod*p*. - Some results on Euler characteristics in combinatorics. I don't really understand how they're proved, or why the "congruent to 1 mod p" comes up.

Looking forward to more situations where "1 mod p" comes up, and/or further insights into the ways I've already mentioned above.