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:
“The difference between objective and subjective extension is one of relation to a context solely.”
—William James (1842–1910)
“Our sympathy is cold to the relation of distant misery.”
—Edward Gibbon (1737–1794)
“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)
“But, most of all, the Great Society is not a safe harbor, a resting place, a final objective, a finished work. It is a challenge constantly renewed, beckoning us toward a destiny where the meaning of our lives matches the marvelous products of our labor.”
—Lyndon Baines Johnson (1908–1973)