Properties
Since each successive factor group Zi+1/Zi in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.
Every subgroup of a nilpotent group of class n is nilpotent of class at most n; in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent of class at most n.
The following statements are equivalent for finite groups, revealing some useful properties of nilpotency:
- G is a nilpotent group.
- If H is a proper subgroup of G, then H is a proper normal subgroup of NG(H) (the normalizer of H in G). This is called the normalizer property and can be phrased simply as "normalizers grow".
- Every maximal proper subgroup of G is normal.
- G is the direct product of its Sylow subgroups.
The last statement can be extended to infinite groups: if G is a nilpotent group, then every Sylow subgroup Gp of G is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup).
Many properties of nilpotent groups are shared by hypercentral groups.
Read more about this topic: Nilpotent Group
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)