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.