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Jordan's derangement theorem: every transitive action of a finite group on a finite set with more than one element has a nonidentity element with no fixed points

Statement

Let a finite group $G$ act transitively on a finite set $X$ with $|X|>1$ . Then some nonidentity $g\in G$ is a derangement:

$X^g=\varnothing.$

Facts & Assumptions

Given: A transitive action of a finite group $G$ on a finite set $X$ with $|X|>1$ .

[L1]

A transitive action has exactly one orbit (Left group actions, transitive actions, and faithful actions).

[L2]

The fixed-point set is $X^g=\{x:g\cdot x=x\}$ (The fixed-point sets $X^g$ and $X^G$ of a group action).

[L3]

Cauchy-Frobenius gives $|G|\,|X/G|=\sum_{g\in G}|X^g|$ (Cauchy-Frobenius orbit counting: $|G|\,|X/G|=\sum_{g\in G}|X^g|$ for a finite group action).

[L4]

Finite cardinalities add over disjoint unions (The sum rule: a finite disjoint union is finite with $\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert$ and $\lvert\bigcup_{i \in I} A_i\rvert = \sum_{i \in I}\lvert A_i\rvert$ , and a sum over a finite index set splits along a partition, The cardinality $\lvert A\rvert$ of a finite set).

[L5]

Finite sums over finite index sets are well-defined (The sum $\sum_{i \in S} a_i$ over a finite index set, and its product form).

Proof

technique · contradiction

1.1

By transitivity [L1], $|X/G|=1$ , so [L3] gives $\sum_{g\in G}|X^g|=|G|$ .

L1L3

1.2

The identity fixes every point, so $|X^e|=|X|$ ; splitting its term from the finite sum gives $\sum_{g\in G}|X^g|=|X|+\sum_{g\ne e}|X^g|$ .

L2L4L5

2.1

Suppose, for contradiction, that every nonidentity $g$ fixes a point. Then every term in the remaining sum is at least $1$ , so step 1.2 gives $\sum_g|X^g|\ge |X|+(|G|-1)>|G|$ , since $|X|>1$ .

assume-contrastep 1.2L2L4L5

3.1

This contradicts step 1.1. Therefore some $g$ has $X^g=\varnothing$ ; the identity fixes all of $X$ , so this $g$ is nonidentity.

step 1.1step 2.1L2discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 76 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Group Actions, Theorem 6.6 (standard reference, not scraped)