# Polycyclic group

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A group possessing a polycyclic series, i.e. a subnormal series with cyclic factors (see Subgroup series). The class of polycyclic groups coincides with the class of solvable groups with the maximum condition for subgroups; it is closed under transition to subgroups, quotient groups and extensions. The number of infinite factors in any polycyclic series is an invariant of the polycyclic group (the polycyclic dimension). The holomorph of a polycyclic group (cf. Holomorph of a group) is isomorphic to a group of matrices over the ring of integers; this enables one to use methods from algebraic geometry, number theory and -adic analysis in the theory of polycyclic groups. If is an algebraic extension of a finite field and is a finite extension of a polycyclic group, then any simple -module is finite-dimensional over . In any group, the product of two locally polycyclic normal subgroups is a locally polycyclic subgroup.