Multiplicative order

The multiplicative order of a group element is the least positive integer n such that its nth power is the identity. It measures the size of the cyclic subgroup generated by that element.

Cyclic subgroup: A subgroup generated by the integer powers of one group element. The element’s order is exactly the number of elements in this subgroup.
Group: A set with an associative operation, an identity element, and an inverse for every element. Multiplicative order is defined for elements of any group, not only numbers.
Primitive root modulo n: An integer whose residue class generates the multiplicative group of units modulo n. A primitive root has order equal to the size of that entire group.
Additive order: The least positive integer n such that n repeated additions of a group element give the identity. It is the same cyclic-length idea expressed with additive notation.
Identity element: The unique group element that leaves every element unchanged under the group operation. The order is the first positive exponent at which the element’s powers reach this element.
Group element: An individual member of a group. The order belongs to an element rather than to the group as a whole.
Discrete logarithm: The problem of finding an exponent x such that g raised to x equals a given group element. The base’s order determines the range and periodicity of possible exponents.
Order of a permutation: The least positive power of a permutation that is the identity permutation. Permutation order is a concrete instance of group-element order.
Lagrange's theorem: A theorem stating that the order of a subgroup of a finite group divides the group’s order. It forces every element’s order in a finite group to divide the group order.
Exponentiation: Repeated application of a multiplication operation, with exponent zero giving the identity. The defining condition asks when repeated powers first return to the identity.

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