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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.
The subgroup = is called a ... is virtually free-by-cyclic, i.e. has a subgroup of finite-index such that there is a free normal subgroup with cyclic quotient /. [26] ...
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.
Here the case of semi-splittings of word-hyperbolic groups over two-ended (virtually infinite cyclic) subgroups was treated by Scott-Swarup [23] and by Bowditch. [24] The case of semi-splittings of finitely generated groups with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of Dunwoody-Swenson. [25]
Here () denotes the classifying space of the group G with respect to the family of virtually cyclic subgroups, i.e. a G-CW-complex whose isotropy groups are virtually cyclic and for any virtually cyclic subgroup of G the fixed point set is contractible. The L-theoretic Farrell–Jones conjecture is analogous.
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.