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 our knowledge and the abyss of being is still real, and the explication must be not less magnificent.”
—Ralph Waldo Emerson (1803–1882)
“What is the vanity of the vainest man compared with the vanity which the most modest person possesses when, in connection with nature and the world, he experiences himself as “man”!”
—Friedrich Nietzsche (1844–1900)
““All men live in suffering
I know as few can know,
Whether they take the upper road
Or stay content on the low....””
—William Butler Yeats (1865–1939)
“The central problem of novel-writing is causality.”
—Jorge Luis Borges (1899–1986)
“Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.”
—Jean Baudrillard (b. 1929)