Connection Between Lower and Upper Central Series
There are various connections between the lower central series and upper central series (Ellis 2001), particularly for nilpotent groups.
Most simply, a group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is trivial) if and only if the UCS stabilizes at the first step (the center is the entire group). 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).
However, the LCS stabilizes at the zeroth step if and only if it is perfect, while the UCS stabilizes at the zeroth step if and only if it is centerless, which are distinct concepts, and show that the lengths of the LCS and UCS need not agree in general.
For a perfect group, the UCS always stabilizes by the first step, a fact called Grün's lemma. However, a centerless group may have a very long lower central series: a noncyclic free group is centerless, but its lower central series does not stabilize until the first infinite ordinal.
Read more about this topic: Central Series
Famous quotes containing the words connection between, connection, upper, central and/or series:
“The connection between dress and war is not far to seek; your finest clothes are those you wear as soldiers.”
—Virginia Woolf (1882–1941)
“The smallest fact about the connection between character and hormonal balance offers more insight into the soul than a five-story idealistic system [of philosophy] does.”
—Robert Musil (1880–1942)
“The thirst for powerful sensations takes the upper hand both over fear and over compassion for the grief of others.”
—Anton Pavlovich Chekhov (1860–1904)
“Parental attitudes have greater correlation with pupil achievement than material home circumstances or variations in school and classroom organization, instructional materials, and particular teaching practices.”
—Children and Their Primary Schools, vol. 1, ch. 3, Central Advisory Council for Education, London (1967)
“Personality is an unbroken series of successful gestures.”
—F. Scott Fitzgerald (1896–1940)