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The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function (sequence A002322 in the OEIS). In other words, λ ( n ) {\displaystyle \lambda (n)} is the smallest number such that for each a coprime to n , a λ ( n ) ≡ 1 ( mod n ) {\displaystyle a^{\lambda (n)}\equiv 1 ...
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p.That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of p n copies of g, and not fewer, is equal to the identity element.
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
If the algebraic group is the multiplicative group mod N, the one-sided identities are recognised by computing greatest common divisors with N, and the result is the p − 1 method. If the algebraic group is the multiplicative group of a quadratic extension of N, the result is the p + 1 method; the calculation involves pairs of numbers modulo N.
n, and is called the group of units modulo n, or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this multiplicative group (× n) is cyclic if and only if n is equal to 2, 4, p k, or 2 p k where p k is a power of an odd prime number.
If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime ≤ k. Proof: Let the AP-k be a·n + b for k consecutive values of n. If a prime p does not divide a, then modular arithmetic says that p will divide every p'th term of the arithmetic ...
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On the other hand, in an infinite cyclic group G = g , the powers g k give distinct elements for all integers k, so that G = { ... , g −2, g −1, e, g, g 2, ... }, and G is isomorphic to the standard group C = C ∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, and the ...