Proof that the order of an element divides the order of the group

Builds on Cyclic subgroup generated by an element Order of a cyclic subgroup equals the order of its generator Lagrange's theorem

Proof

Let $G$ be a finite group and let $g \in G$ .
The cyclic subgroup $\langle g \rangle$ is a subgroup of $G$ .
The order of $\langle g \rangle$ is the order of $g$ .
By Lagrange's theorem, $|\langle g \rangle|$ divides $|G|$ .
Thus the order of $g$ divides $|G|$ .

Idea

The element generates a subgroup whose order is the order of the element. Lagrange's theorem then gives the result.

Reading path

To understand this page from the ground up, read these 42 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
Laws of exponents in a group statement
Order of a group definition
Subgroup 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
Cyclic subgroup generated by an element definition
Left coset definition
Right coset definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Disjoint sets definition
g to the n equals 1 iff the order divides n statement
Subset definition
Union of a family of sets definition
Cosets have the same cardinality as the subgroup statement
Binary relation definition
Partition of a set definition
Order of a cyclic subgroup equals the order of its generator statement
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Equivalence relation definition
Equivalence class definition
Equivalence classes form a partition statement
Proof of Lagrange's theorem proof
Lagrange's theorem statement
Proof that the order of an element divides the order of the group proof

Where this leads

4 pages build on this one.

Tag	`BMD35Z5SGC`
URI	`m2://BMD35Z5SGC`
Kind	proof
Writing	W0 stub
MSC	20D60
Proves	Order of an element divides the order of the group
Uses	Cyclic subgroup generated by an element, Order of a cyclic subgroup equals the order of its generator, Lagrange's theorem
Methods	Direct proof

Proof that the order of an element divides the order of the group

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here