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  2. Cyclic number - Wikipedia

    en.wikipedia.org/wiki/Cyclic_number

    where b is the number base (10 for decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers in base b are called full reptend primes or long primes in base b). For example, the case b = 10, p = 7 gives the cyclic number 142857, and the case b = 12, p = 5 gives the cyclic number 2497.

  3. Cyclic number (group theory) - Wikipedia

    en.wikipedia.org/wiki/Cyclic_number_(group_theory)

    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 …

  4. Cycles and fixed points - Wikipedia

    en.wikipedia.org/wiki/Cycles_and_fixed_points

    (3.a) If we want element k to be a fixed point we may choose one of the s(k − 1, j − 1) permutations with k − 1 elements and j − 1 cycles and add element k as a new cycle of length 1. (3.b) If we want element k not to be a fixed point we may choose one of the s(k − 1, j) permutations with k − 1 elements and j cycles and insert ...

  5. Cyclic group - Wikipedia

    en.wikipedia.org/wiki/Cyclic_group

    If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. 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 ...

  6. Full reptend prime - Wikipedia

    en.wikipedia.org/wiki/Full_reptend_prime

    The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime. That is, the multiplicative order ord p b = p − 1, which is equivalent to b being a primitive root modulo p. The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers.

  7. Lagrange's theorem (group theory) - Wikipedia

    en.wikipedia.org/wiki/Lagrange's_theorem_(group...

    Since any element of the form (a b c) squared is (a c b), and (a b c)(a c b) = e, any element of H in the form (a b c) must be paired with its inverse. Specifically, the remaining 5 elements of H must come from distinct pairs of elements in A 4 that are not in V. This is impossible since pairs of elements must be even and cannot total up to 5 ...

  8. Difference set - Wikipedia

    en.wikipedia.org/wiki/Difference_set

    In combinatorics, a (,,) difference set is a subset of size of a group of order such that every non-identity element of can be expressed as a product of elements of in exactly ways. A difference set D {\displaystyle D} is said to be cyclic , abelian , non-abelian , etc., if the group G {\displaystyle G} has the corresponding property.

  9. Finite group - Wikipedia

    en.wikipedia.org/wiki/Finite_group

    A cyclic group Z n is a group all of whose elements are powers of a particular element a where a n = a 0 = e, the identity. A typical realization of this group is as the complex n th roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.