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Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; [note 3] an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n. Every abelian subgroup of a Gromov hyperbolic group is ...
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 ...
G 0 is the trivial subgroup; G i is a normal subgroup of G i+1 (for every i between 0 and n - 1) and the quotient group G i+1 / G i is a cyclic group (for every i between 0 and n - 1) A metacyclic group is a polycyclic group with n ≤ 2, or in other words an extension of a cyclic group by a cyclic group.
Any finite group (since the trivial subgroup is the free group on the empty set of generators). Any virtually cyclic group. (Either it is finite in which case it falls into the above case, or it is infinite and contains as a subgroup.) Any semidirect product where N is free and H is finite.
Elements of the representation ring are sometimes called virtual representations. [1] For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed fields of characteristic p where the Sylow p -subgroups are cyclic is also ...
Here, each () / is a cyclic subgroup of Z(p ∞) with p n elements; it contains precisely those elements of Z(p ∞) whose order divides p n and corresponds to the set of p n-th roots of unity. The Prüfer p -groups are the only infinite groups whose subgroups are totally ordered by inclusion.
If F ≤ G is a finite subgroup then F is conjugate in G to a subgroup of some vertex group A v. If the graph A is finite and all vertex groups A v are finite then the group G is virtually free, that is, G contains a free subgroup of finite index. If A is finite and all the vertex groups A v are finitely generated then G is finitely generated.
The group SL 2 (Z) is not boundedly generated, since it contains a free subgroup with two generators of index 12. A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or elementary), i.e. contains a cyclic subgroup of finite index.