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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. 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 ...

  5. 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.

  6. List of finite simple groups - Wikipedia

    en.wikipedia.org/wiki/List_of_finite_simple_groups

    The outer automorphism group is often, but not always, isomorphic to the semidirect product () where all these groups ,, are cyclic of the respective orders d, f, g, except for type (), odd, where the group of order = is , and (only when =) =, the symmetric group on three elements.

  7. Regular p-group - Wikipedia

    en.wikipedia.org/wiki/Regular_p-group

    A 3-group with two generators is regular if and only if its derived subgroup is cyclic. Every p-group of odd order with cyclic derived subgroup is regular. The subgroup of a p-group G generated by the elements of order dividing p k is denoted Ω k (G) and regular groups are well-behaved in that Ω k (G) is precisely the set of elements of order ...

  8. 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 ...

  9. Automorphic number - Wikipedia

    en.wikipedia.org/wiki/Automorphic_number

    Given a number base , a natural number with digits is an automorphic number if is a fixed point of the polynomial function = over /, the ring of integers modulo.As the inverse limit of / is , the ring of -adic integers, automorphic numbers are used to find the numerical representations of the fixed points of () = over .