Lagrange's theorem ↩

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created: 2021-12-06 12:59:59
modified: 2022-01-10 04:13:04

Statement: The order of a finite group is always a multiple of the order of its subgroups.

Proof

Since cosets partition the group into equal partitions and the order of each coset equals the order of its subgroup, the order of the subgroup divides the order of the group.

Corollaries

The order of an element in a finite group divides the order of the group
Groups of prime order cannot have any nontrivial subgroups
Every element of a group of prime order has an order equal to the order of the group, except for the neutral element itself (whose order is 1)