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Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...
A cyclic number [1] [2] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. [3] Any prime number is clearly cyclic. All cyclic numbers are square-free. [4] Let n = p 1 p 2 …
A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are:
Cyclic number, a number such that cyclic permutations of the digits are successive multiples of the number; Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle; Cyclic permutation, a permutation with one nontrivial orbit; Cyclic polygon, a polygon which can be given a circumscribed circle
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. [1] [2] In some cases, cyclic permutations are referred to as cycles; [3] if a cyclic permutation has k elements, it may be called a k-cycle. Some authors widen this definition to include permutations with fixed points in ...
The cyclic group = (/, +) = of congruence classes modulo 3 (see modular arithmetic) is simple.If is a subgroup of this group, its order (the number of elements) must be a divisor of the order of which is 3.
In mathematics, especially group theory, ... The class number of is the number ... A cyclic permutation of three (other one remains unchanged). ...
Furthermore, if is the smallest prime number dividing the order of a finite group, , then if / has order , must be a normal subgroup of . [ 3 ] Given G {\displaystyle G} and a normal subgroup N {\displaystyle N} , then G {\displaystyle G} is a group extension of G / N {\displaystyle G\,/\,N} by N {\displaystyle ...