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:
“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)
“Every word was once a poem. Every new relation is a new word.”
—Ralph Waldo Emerson (1803–1882)
“Authority and power are two different things: power is the force by means of which you can oblige others to obey you. Authority is the right to direct and command, to be listened to or obeyed by others. Authority requests power. Power without authority is tyranny.”
—Jacques Maritain (1882–1973)
“Isn’t it odd that networks accept billions of dollars from advertisers to teach people to use products and then proclaim that children aren’t learning about violence from their steady diet of it on television!”
—Toni Liebman (20th century)