Cyclic group and Prime power

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Difference between Cyclic group and Prime power

Cyclic group vs. Prime power

In algebra, a cyclic group or monogenous group is a group that is generated by a single element. In mathematics, a prime power is a positive integer power of a single prime number.

Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

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Exponentiation

Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Multiplicative group of integers modulo n

In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Primitive root modulo n

In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n).

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

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