nilpotent group sentence in Hindi
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- 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.
- 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.
- 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 ).
- In particular any unipotent group is a nilpotent group, though the converse is not true ( counterexample : the diagonal matrices of GL " n " ( " k " ) ).
- 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 ).
- 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 ).
- 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.
- 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.
- 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
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 ).
- 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 .)
- 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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