Relation To Direct Products
Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism G → N which is the identity on N, then G is the direct product of N and H.
The direct product of two groups N and H can be thought of as the outer semidirect product of N and H with respect to φ(h) = idN for all h in H.
Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles.
Read more about this topic: Semidirect Product
Famous quotes containing the words relation to, relation, direct and/or products:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“You must realize that I was suffering from love and I knew him as intimately as I knew my own image in a mirror. In other words, I knew him only in relation to myself.”
—Angela Carter (1940–1992)
“A temple, you know, was anciently “an open place without a roof,” whose walls served merely to shut out the world and direct the mind toward heaven; but a modern meeting-house shuts out the heavens, while it crowds the world into still closer quarters.”
—Henry David Thoreau (1817–1862)
“All that is told of the sea has a fabulous sound to an inhabitant of the land, and all its products have a certain fabulous quality, as if they belonged to another planet, from seaweed to a sailor’s yarn, or a fish story. In this element the animal and vegetable kingdoms meet and are strangely mingled.”
—Henry David Thoreau (1817–1862)