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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 ...
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 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 ...
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]
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 () 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.
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 ...
One of the consequences of Bowditch's work is that for one-ended word-hyperbolic groups (with a few exceptions) having a nontrivial essential splitting over a virtually cyclic subgroup is a quasi-isometry invariant. Bowditch also gave a topological characterisation of word-hyperbolic groups, thus solving a conjecture proposed by Mikhail Gromov.