Suppose G is a nilpotent group of nilpotency class c_1. We can calculate the c_2-nilpotent multiplier of G as follows. Let F be a free nilpotent ... ...
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It is possible to have a nilpotent group G such that G cannot be embedded ... | nilpotent group || none || divisible group || No || nilpotent group ... ...
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Suppose G is the free nilpotent group of class c_1 on a generating ... second half of lower central series of nilpotent group comprises abelian groups ... ...
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numbers and G is a \pi-torsion-free nilpotent group, i.e., G is a nilpotent ... G such that K is a \pi-powered nilpotent group and such that for any ... ...
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Suppose G is a nilpotent group of nilpotency class c. We can calculate ... Suppose G is a nilpotent group of nilpotency class c. We can calculate ... ...
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#REDIRECT Variant of Hopf's formula for nilpotent multiplier of nilpotent group that uses the free nilpotent group of class equal to the sum ... ...
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It is possible to have a free nilpotent group (in fact, we can choose ... that the derived subgroup is not a free nilpotent group. ... ...
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Suppose \pi is a set of primes and G is a \pi-powered nilpotent group ... for Schur multiplier of powered nilpotent group, and is similar to how ... ...
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or equal to c. Denote by G the free pi-powered nilpotent group of class c on a set S and denote by L the free pi-powered nilpotent Lie ring on S. ... ...
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Equivalence of definitions of nilpotent group, equivalence of definitions ... 8 is the smallest (in terms of order) nilpotent group which is not abelian ... ...
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#REDIRECT Group whose automorphism group is nilpotent ... ...
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groups. Suppose further that G_1 is a finite nilpotent group. Then, G_1 and G_2 are isomorphic groups. In particular, G_2 is also a finite nilpotent group. ... ...
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A finite group is termed a finite nilpotent group if it satisfies ... # It is a nilpotent group.
# It satisfies the normalizer condition ... ...
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nilpotent group
==Definition==
A group is said to be locally nilpotent ... * Any nilpotent group, including any abelian group or any finite nilpotent ... ...
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residually operator|nilpotent group
==Relation with other properties== ... | Weaker than::nilpotent group || lower central series reaches identity ... ...
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A periodic nilpotent group or locally finite nilpotent group is defined ... # It is a nilpotent group and also a group generated by periodic elements: ... ...
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A finite group G is termed a p-nilpotent group for a prime number p if the following equivalent conditions are satisfied:
# G has a defining ingredient::normal ... ...
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virtually operator|nilpotent group
==Relation with other properties==
===Stronger properties===
* Weaker than::Virtually abelian group ... ...
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#redirect finitely generated nilpotent group ... ...
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Note that this differs from the notion of p-nilpotent group. Thus, when somebody uses n-nilpotent for a specific numerical value of n that is ... ...
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#REDIRECT Variant of Hopf's formula for nilpotent multiplier of nilpotent group that uses the free nilpotent group of class equal to the sum ... ...
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* Nilpotent group
* Nilpotent Lie ring ... ...
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The automorphism group of a nilpotent group need not be nilpotent ... group, which is an abelian and hence nilpotent group, is the symmetric group ... ...
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==Statement==
Let H be a finite nilpotent homomorphism-closed subgroup of a group G. Then, H must be equal to G.
A variant of this is the claim ... ...
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An automorphism of a nilpotent group is said to be nilpotent-extensible ... Let G be a nilpotent group and \sigma an automorphism of G. Then we ... ...
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A group is termed a rationally powered nilpotent group if it satisfies ... # It is both a rationally powered group and a nilpotent group. ... ...
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A nilpotent group in which every automorphism is inner is a group ... an infinite (and infinitely generated) nilpotent group. See finitely generated ... ...
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* The group is nilpotent, i.e., it is a finite nilpotent group. by any two elements of the group is a nilpotent group.
===Statement ... ...
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In a nilpotent group, every subgroup is subnormal. Moreover, the fact ... every subgroup of a residually nilpotent group is descendant. ... ...
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A group is termed a 2-locally nilpotent group if every 2-generated ... The 2-local nilpotency class of a 2-locally nilpotent group is defined ... ...
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Let G be a nilpotent p-group, i.e., a nilpotent group where the order of every element is a power of the prime p. Then, the subgroup \Omega_1 ... ...
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We say that G is a \pi-divisible nilpotent group if it satisfies the ... Equivalence of definitions of nilpotent group that is divisible for ... ...
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characteristic subgroup|nilpotent group
==Definition== The term characteristic subgroup of nilpotent group is used for a ... ...
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normal subgroup|locally nilpotent group
==Definition==
A subgroup of a group is termed a locally nilpotent normal subgroup if it is locally nilpotent ... ...
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2-powered group|nilpotent group
==Definition== A group is termed a 2-powered nilpotent group if it is both a 2-powered ... ...
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the quotient is a defining ingredient::nilpotent group). ... When the group is a finite group or a virtually nilpotent group, and ... ...
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For a fact about::nilpotent group, the fact about::commutator subgroup ... # Nilpotent implies every maximal subgroup is normal: In a nilpotent ... ...
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virtually operator|nilpotent group
==Relation with other properties==
===Stronger properties===
* Weaker than::Virtually abelian group ... ...
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stronger = locally nilpotent group|
weaker = nilpotent group}} ... gives an example of a locally nilpotent group that does not satisfy ... ...
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# Every group in the variety is a nilpotent group
# There exists a constant c such that every group in the variety is nilpotent with nilpotence ... ...
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