Lagrange's theorem

In a finite group, the order of every subgroup divides the order of the group. Equivalently, the group is partitioned into equally sized cosets of that subgroup.

Coset: A coset is a translate of a subgroup within a group. Every left coset has the subgroup’s size, producing the equal-sized partition behind the theorem.
Group (mathematics): A group is a set with an associative operation, an identity, and inverses for every element. Lagrange’s theorem is a statement about the sizes of groups and their subgroups.
Fermat's little theorem: For prime p and integer a not divisible by p, Fermat’s little theorem states that a^(p−1) is congruent to 1 modulo p. It follows by applying Lagrange’s theorem to the multiplicative group modulo p.
Cyclic group: A cyclic group is generated by a single element. Every divisor of a finite cyclic group’s order occurs as a subgroup order, unlike in general groups.
Index of a subgroup: The index of a subgroup is the number of its left cosets in a group. The group order equals subgroup order multiplied by this coset count.
Subgroup: A subgroup is a subset of a group that forms a group under the same operation. The theorem relates each subgroup’s size to the size of its containing group.
Euler's theorem: Euler’s theorem states that a^φ(n) is congruent to 1 modulo n when a and n are coprime. The exponent φ(n) is the order of the finite group of invertible residues modulo n.
Subgroup lattice: A subgroup lattice is the partially ordered set of a group’s subgroups, ordered by inclusion. Lagrange’s divisibility condition restricts possible subgroup sizes without determining the lattice.
Group action: A group action is a way a group acts as symmetries of a set. Orbit-stabilizer generalizes the same divisibility argument from cosets to actions.
Order (group theory): The order of a group is its number of elements; the order of an element is its least positive identity-producing power. The theorem compares group orders, and its corollaries constrain element orders.

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