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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

    [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.

  4. Group isomorphism - Wikipedia

    en.wikipedia.org/wiki/Group_isomorphism

    From an algebraic point of view, this means that the set of all integers (with the addition operation) is the "only" infinite cyclic group. Some groups can be proven to be isomorphic, relying on the axiom of choice, but the proof does not indicate how to construct a concrete isomorphism. Examples:

  5. Presentation of a group - Wikipedia

    en.wikipedia.org/wiki/Presentation_of_a_group

    Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the presentation = , where 1 is the group identity.

  6. Finiteness properties of groups - Wikipedia

    en.wikipedia.org/.../Finiteness_properties_of_groups

    In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely ...

  7. Cyclic module - Wikipedia

    en.wikipedia.org/wiki/Cyclic_module

    2Z as a Z-module is a cyclic module. In fact, every cyclic group is a cyclic Z-module. Every simple R-module M is a cyclic module since the submodule generated by any non-zero element x of M is necessarily the whole module M. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements. [2]

  8. Locally cyclic group - Wikipedia

    en.wikipedia.org/wiki/Locally_cyclic_group

    The additive group of rational numbers (Q, +) is locally cyclic – any pair of rational numbers a/b and c/d is contained in the cyclic subgroup generated by 1/(bd). [2]The additive group of the dyadic rational numbers, the rational numbers of the form a/2 b, is also locally cyclic – any pair of dyadic rational numbers a/2 b and c/2 d is contained in the cyclic subgroup generated by 1/2 max ...

  9. Free-by-cyclic group - Wikipedia

    en.wikipedia.org/wiki/Free-by-cyclic_group

    In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group is said to be free-by-cyclic if it has a free normal subgroup such that the quotient group / is cyclic. In other words, is free-by-cyclic if it can be expressed as a group extension of a free group by