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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. List of small groups - Wikipedia

    en.wikipedia.org/wiki/List_of_small_groups

    those of order p 7 for p = 3, 5, 7, 11 (907 489 groups); those of order pq n where q n divides 2 8, 3 6, 5 5 or 7 4 and p is an arbitrary prime which differs from q; those whose orders factorise into at most 3 primes (not necessarily distinct). It contains explicit descriptions of the available groups in computer readable format.

  4. List of finite simple groups - Wikipedia

    en.wikipedia.org/wiki/List_of_finite_simple_groups

    Outer automorphism group: Cyclic of order p − 1. ... Conway group, Co 3; Order 2 21 ⋅ 3 9 ⋅ 5 4 ⋅ 7 2 ⋅ 11 ⋅ 13 ⋅ 23 = 4157776806543360000

  5. Subgroups of cyclic groups - Wikipedia

    en.wikipedia.org/wiki/Subgroups_of_cyclic_groups

    [5] [6] [7] (See also cyclic group for some characterization.) There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example. However, the Klein group has more than one subgroup of order 2, so it does not meet the conditions of the characterization.

  6. Classification of finite simple groups - Wikipedia

    en.wikipedia.org/wiki/Classification_of_finite...

    In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...

  7. Small Latin squares and quasigroups - Wikipedia

    en.wikipedia.org/wiki/Small_Latin_squares_and...

    There are two main classes and only two isotopy classes, but 1,411 isomorphism classes. There are six isomorphism classes that contain reduced squares, that is, there are six loops, only one of which is a group, the cyclic group of order five. [1] Below are two reduced Latin squares of order five, one from each isotopy class. [10]

  8. Order (group theory) - Wikipedia

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

    The order of a group G is denoted by ord(G) or | G |, and the order of an element a is denoted by ord(a) or | a |, instead of ⁡ ( ), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup H of a finite group G , the order of the subgroup divides the order of the group; that is, | H | is a divisor of | G | .

  9. Cyclically ordered group - Wikipedia

    en.wikipedia.org/wiki/Cyclically_ordered_group

    Given a cyclically ordered group K and an ordered group L, the product K × L is a cyclically ordered group. In particular, if T is the circle group and L is an ordered group, then any subgroup of T × L is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with T. [3]