Fixed-point congruence for $p$-groups

A finite $p$-group action has total points congruent mod $p$ to its fixed points.Suggest a slogan

Let $p$ be a prime, let $P$ be a finite group of order $p^a$, and let $P$ act on a finite set $X$, with fixed-point set $X^P=\theset{x\in X\suchthat g\cdot x=x\text{ for every }g\in P}$. Then \[ \abs X\equiv\abs{X^P}\pmod p. \]

Partition $X$ into its $P$-orbits. A fixed point contributes an orbit of size one. Every other orbit $P\cdot x$ has size $[P:P_x]$ by the orbit-stabilizer theorem, which is a power of $p$ larger than one. Those nontrivial orbit sizes vanish modulo $p$, leaving only the fixed points.

Guide Appearances

Group actions and counting

Dependencies

D-WYC7CGroup actions, orbits, and stabilizersuses
T-QYDVHOrbit-stabilizer theoremuses

Backlinks

T-WRMBMSylow's first theorem: existence of prime-power subgroupsuses
T-EF2MZSylow's second theorem: Sylow subgroups are conjugateuses
T-4CDTTThe class equation and centers of finite $p$-groupsuses