Lagrange theorem on subgroup size

Lagrange’s Theorem states that if $$G$$ is a finite group and $$H$$ a subgroup, then the order $$|H|$$ of $$H$$ divides the order $$|G|$$ of $$G$$. It generalises to infinite groups: the statement then becomes that the left cosets form a partition, and for any pair of cosets, there is a bijection between them.

Proof

In full generality, the cosets form a partition and are all in bijection.

To specialise this to the finite case, we have divided the $$|G|$$ elements of $$G$$ into buckets of size $$|H|$$ (namely, the cosets), so $$|G|/|H|$$ must in particular be an integer.

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• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.