Group theory and generalizations Finite groups Theorem

Cauchy's theorem

Builds on Order of an element Lagrange's theorem Fixed points of a p-group action

Statement

Let $G$ be a finite group and let $p$ be a prime number that divides $|G|$ . Then $G$ has an element of order $p$ .

Remarks

Equivalently, $G$ has a subgroup of order $p$ , namely the cyclic subgroup generated by such an element. It is a partial converse of Lagrange's theorem: a prime divisor of $|G|$ is always the order of a subgroup. One proof lets $\mathbb{Z}/p\mathbb{Z}$ act by cyclic shifts on the set of tuples $(g_1, \dots, g_p)$ with $g_1 g_2 \cdots g_p = e$ , and uses the fixed point lemma. The first Sylow theorem generalises the result to prime powers.

History

Augustin-Louis Cauchy discovered the theorem in 1845.

Reading path

To understand this page from the ground up, read these 60 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

Element definition
Group concept
Left coset concept
Proposition concept
Set definition
Conjugate elements definition
Group homomorphism definition
Laws of exponents in a group statement
Order of a group definition
Subgroup definition
Symmetric group definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Order of an element definition
Intersection definition
Union definition
Conjugate subgroup definition
Group action definition
Left coset definition
Right coset definition
Set-builder notation concept
Subgroup criterion statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Disjoint sets definition
P-group definition
Normal subgroup definition
Subset definition
Stabiliser definition
Union of a family of sets definition
Orbit definition
Index of a subgroup definition
Intersection of subgroups is a subgroup statement
Cosets have the same cardinality as the subgroup statement
Binary relation definition
Partition of a set definition
Normal core definition
Fixed points of an action definition
Orbits partition the set statement
Transitive action definition
Order of a group is the index times the order of the subgroup statement
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Action on left cosets example
Equivalence relation definition
Orbit-stabiliser theorem statement
Equivalence class definition
Equivalence classes form a partition statement
Proof of Lagrange's theorem proof
Lagrange's theorem statement
Size of an orbit times the size of the stabiliser statement
Fixed points of a p-group action statement
Cauchy's theorem statement

Where this leads

13 pages build on this one.

Tag	`0JM7PA288B`
URI	`m2://0JM7PA288B`
Aliases	`cauchys-theorem-group-theory`
Kind	statement, theorem
Writing	W1 stated
MSC	20D20
Status	This copy of the site has no build data. Run `m2 status 0JM7PA288B`.
Uses	Order of an element, Lagrange's theorem, Fixed points of a p-group action
Same as	Wikidata Q1139041 Wikipedia Cauchy's theorem (group theory)

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