Definition
This motivates the definition of the commutator subgroup (also called the derived subgroup, and denoted G′ or G(1)) of G: it is the subgroup generated by all the commutators.
It follows from the properties of commutators that any element of is of the form
for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,
- ,
so that .
This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below. Moreover, taking G = H it shows that the commutator subgroup is stable under every endomorphism of G: that is, is a fully characteristic subgroup of G, a property which is considerably stronger than normality.
The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g = g1 g2 ... gk that can be rearranged to give the identity.
Read more about this topic: Commutator Subgroup
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)