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)
“The problem of the twentieth century is the problem of the color-line—the relation of the darker to the lighter races of men in Asia and Africa, in America and the islands of the sea. It was a phase of this problem that caused the Civil War.”
—W.E.B. (William Edward Burghardt)
“As for your friend, my prospective reader, I hope he ignores Fort Sumter, and “Old Abe,” and all that; for that is just the most fatal, and, indeed, the only fatal weapon you can direct against evil ever; for, as long as you know of it, you are particeps criminis. What business have you, if you are an “angel of light,” to be pondering over the deeds of darkness, reading the New York Herald, and the like.”
—Henry David Thoreau (1817–1862)
“... white people, like black ones, are victims of a racist society. They are products of their time and place.”
—Shirley Chisholm (b. 1924)