UL-equivalent group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
Symbol-free definition
A group is said to be UL-equivalentif it is nilpotent and its upper central series and lower central series actually coincide. In other words, if the nilpotence class is , the
term of the Upper central series (?) equals the
term of the Lower central series (?). (Note that we start counting the lower central series from
at the group, and the upper central series at
from the trivial subgroup).
Definition with symbols
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Abelian group | Nilpotency class is 0 or 1 | abelian implies UL-equivalent | UL-equivalent not implies abelian (see also list of examples) | |FULL LIST, MORE INFO |
Special group | Group of prime power order whose center, derived subgroup, and Frattini subgroup coincide (according to some conventions, we also add in the elementary abelian groups) | |FULL LIST, MORE INFO | ||
Extraspecial group | Special group where the center is cyclic | (via special) | (via special) | |FULL LIST, MORE INFO |
Maximal class group | Group of order ![]() ![]() ![]() ![]() |
maximal class implies UL-equivalent | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Nilpotent group | (by definition) | nilpotent not implies UL-equivalent (see also list of examples) | |FULL LIST, MORE INFO |