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  2. Cyclic group - Wikipedia

    en.wikipedia.org/wiki/Cyclic_group

    A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.

  3. Subgroups of cyclic groups - Wikipedia

    en.wikipedia.org/wiki/Subgroups_of_cyclic_groups

    The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a divisor ...

  4. p-group - Wikipedia

    en.wikipedia.org/wiki/P-group

    The trivial group is the only group of order one, and the cyclic group C p is the only group of order p. There are exactly two groups of order p 2, both abelian, namely C p 2 and C p × C p. For example, the cyclic group C 4 and the Klein four-group V 4 which is C 2 × C 2 are both 2-groups of order 4.

  5. Multiplicative group of integers modulo n - Wikipedia

    en.wikipedia.org/wiki/Multiplicative_group_of...

    By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers ,,, …, give all possible residues modulo n coprime to n (the first () powers , …, give each exactly once).

  6. Quaternion group - Wikipedia

    en.wikipedia.org/wiki/Quaternion_group

    It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. [12] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. [13]

  7. Quotient group - Wikipedia

    en.wikipedia.org/wiki/Quotient_group

    One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group / is the group of three colors, which turns out to be the cyclic group with three elements.

  8. Alternating group - Wikipedia

    en.wikipedia.org/wiki/Alternating_group

    The Schur multipliers of the alternating groups A n (in the case where n is at least 5) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6. [3] These were first computed in .

  9. 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 …