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:
“Its 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)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than thisdevoted and obedient. This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (18201910)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)