Tour:Lagrange's theorem

This article adapts material from the main article: Lagrange's theorem

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Statement

In a finite group, the order of any subgroup divides the order of the group. In fact, the ratio of the orders is the index of the subgroup, which is the same as the number of left cosets, and as the number of right cosets, of the subgroup.

Proof

Key ingredient

The key ingredient for the proof is the fact that left cosets are in bijection via left multiplication. This establishes that left cosets all have the same cardinality, and hence the group is partitioned into as many copies of the subgroup as the number of left cosets. This establishes Lagrange's theorem combinatorially.