Since each successive factor group " Z " " i " + 1 / " Z " " i " in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.
32.
The result obtained is actually a bit stronger since it establishes that there exists a " growth gap " between virtually nilpotent groups ( of polynomial growth ) and other groups; that is, there exists a ( superpolynomial ) function f such that any group with growth function bounded by a multiple of f is virtually nilpotent.
33.
If " ? " is the first infinite ordinal, then " G " ? is the smallest normal subgroup of " G " such that the quotient is "'residually nilpotent "', that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group.
34.
In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the " Q "-type have 2 classes of involutions and one class of cyclic subgroup of order 4; the " QD "-type have one class each of involutions and cyclic subgroups of order 4.
35.
In the language of formations, a Sylow " p "-subgroup is a covering group for the formation of " p "-groups, a Hall " ? "-subgroup is a covering group for the formation of " ? "-groups, and a Carter subgroup is a covering group for the formation of nilpotent groups.
36.
The last statement can be extended to infinite groups : if " G " is a nilpotent group, then every Sylow subgroup " G " " p " of " G " is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in " G " ( see torsion subgroup ).
37.
For a nilpotent group, the smallest n such that G has a central series of length n is called the "'nilpotency class "'of G; and G is said to be "'nilpotent of class n "'. ( By definition, the length is n if there are n + 1 different subgroups in the series, including the trivial subgroup and the whole group .)
38.
During the course of the Alperin Brauer Gorenstein theorem classifying finite simple groups with quasi-dihedral Sylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups : the 2-nilpotent groups, the " Q "-type groups whose focal subgroup is a generalized quaternion group of index 2, the " D "-type groups whose focal subgroup a dihedral group of index 2, and the " QD "-type groups whose focal subgroup is the entire quasi-dihedral group.
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