More precisely, in a nilpotent group satisfying this condition lattices correspond via the exponential map to lattices ( in the more elementary sense of Lattice ( group ) ) in the Lie algebra.
22.
Finally, a nilpotent group is isomorphic to a lattice in a nilpotent Lie group if and only if it contains a subgroup of finite index which is torsion-free and of finitely generated.
23.
The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization : every homogeneous nilmanifold is isometric to a nilpotent Lie group with left-invariant metric ( see Wilson ).
24.
In particular any unipotent group is a nilpotent group, though the converse is not true ( counterexample : the diagonal matrices of GL " n " ( " k " ) ).
25.
More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree ( and is called the "'nilpotency class "'of the group ).
26.
In the case of a nilpotent group " G " the correspondence involves all orbits, but for a general " G " additional restrictions on the orbit are necessary ( polarizability, integrality, Pukanszky condition ).
27.
An ingredient of the proof of Brauer's induction theorem is that when " G " is a finite nilpotent group, every complex irreducible character of " G " is induced from a linear character of some subgroup.
28.
Hypercentral groups enjoy many properties of nilpotent groups, such as the "'normalizer condition "'( the normalizer of a proper subgroup properly contains the subgroup ), elements of coprime order commute, and Sylow " p "-subgroups.
29.
Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line \ Re s = 1, they showed that for any torsionfree nilpotent group, the function ? " G " ( " s " ) is meromorphic in the domain
30.
This is not a defining characteristic of nilpotent groups : groups for which \ operatorname { ad } _ g is nilpotent of degree " n " ( in the sense above ) are called " n "-Engel groups, and need not be nilpotent in general.
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