UL-equivalent group

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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 $c$, the $k^{th}$ term of the Upper central series (?) equals the $(c+1-k)^{th}$ term of the Lower central series (?). (Note that we start counting the lower central series from $1$ at the group, and the upper central series at $0$ from the trivial subgroup).

Definition with symbols

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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 $p^n$, nilpotency class $n - 1$, with $n \ge 3$, $p$ prime	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