If a finite $p$-group $P$ acts on a finite set $X$, then $\mathrm{\mid }X\mathrm{\mid }\equiv \mathrm{\mid }{X}^P\mathrm{\mid }(\mathrm{m}\mathrm{o}\mathrm{d}p)$

A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$

The fixed-point sets $X^g$ and $X^G$ of a group action

The orbits of a group action are the equivalence classes of $x\sim y$ iff $y=g\cdot x$ for some $g$, and hence partition the acted-on set

Orbit-stabiliser cardinality: $\mathrm{\mid }G\cdot x\mathrm{\mid }=[G:G_x]$ whenever either side is finite, and $\mathrm{\mid }G\mathrm{\mid }=\mathrm{\mid }{G}_x\mathrm{\mid }\mathrm{\mid }G\cdot x\mathrm{\mid }$ for finite $G$

Every subgroup of a finite $p$-group has order a power of $p$

Congruence modulo an integer: $a\equiv b(\mathrm{m}\mathrm{o}\mathrm{d}n)$ when $n\mid (a-b)$, including the moduli $0$ and $1$

The sum rule: a finite disjoint union is finite with $\mid A\cup B\mid =\mid A\mid +\mid B\mid$ and $\mid {\bigcup }_{i\in I}{A}_i\mid ={\sum }_{i\in I}\mid A_i\mid$, and a sum over a finite index set splits along a partition

The sum ${\sum }_{i\in S}{a}_i$ over a finite index set, and its product form