Carmichael's theorem

Every finite subgroup of the multiplicative group of a field is cyclic. Equivalently, any finite field’s nonzero elements form a cyclic group under multiplication.

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Lagrange's theorem: In a finite group, the order of each subgroup divides the order of the group. It constrains the possible orders of elements in the finite subgroup.
Field (mathematics): A set with addition and multiplication satisfying the field axioms, including multiplicative inverses for nonzero elements. The theorem applies to multiplicative subgroups drawn from fields.
Finite field: A field with finitely many elements. Its nonzero elements form a cyclic group by the theorem.
Finite abelian group: A finite group in which every pair of elements commutes. The theorem gives a stronger conclusion than commutativity for these particular field subgroups.
Order of a group element: The least positive exponent that raises a group element to the identity. Carmichael’s theorem is proved by analyzing the orders of subgroup elements.
Multiplicative group of a field: The group of a field’s nonzero elements under multiplication. Finite subgroups of this group are exactly the objects the theorem classifies.
Finite field multiplicative group: The group of nonzero elements of a finite field under multiplication. The theorem guarantees a generator for this group, enabling exponent-based descriptions.
Klein four-group: The noncyclic group of order four in which every nonidentity element has order two. It is a finite abelian group, but cannot embed in a field’s multiplicative group.
Primitive element: An element that generates a specified cyclic group or field extension. A generator of the finite subgroup is precisely a primitive element of that group.
Finite group: A group containing finitely many elements. The theorem concerns finite subgroups, whose possible structures are sharply constrained.