Definition
The definition uses the idea, explained on its own page, of a central series for a group. The following are equivalent formulations:
- A nilpotent group is one that has a central series of finite length.
- A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps.
- A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps.
For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G ; and G is said to be nilpotent of class n. (By definition, the length is n if there are n + 1 different subgroups in the series, including the trivial subgroup and the whole group.)
Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most m, then it is sometimes called a nil-m group.
It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups.
Read more about this topic: Nilpotent Group
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)