Lagrange's theorem (group theory)

The set of Lagrange is a mathematical theorem of group theory, which states that the cardinality (or order ) of each subgroup of a finite group divides their thickness. It was named after the Italian mathematician Joseph -Louis Lagrange.

Statement

Let be a finite group, then:

For a finite group, the set of Lagrange is accurate:

Using as the index, which is the number of sub- classes from to. Since the index is an integer, it also follows that a divisor of is.

Proof of the theorem

Be subgroup of the finite group.

Consider for each the (left ) coset.

There is a bijection between and since. Therefore, all cosets are equal ( and have elements).

If two cosets, one element in common, so the cosets are even equal.

Finally, since quite cover the cosets ( it is), is broken down many disjoint - element subsets in finite, so that must be a multiple of.

The second statement of the theorem is a simple consequence of the first, as that currently has the cardinality of generated subgroup.