Elementary Facts and Caveats
If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H.
Note that, as opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if G and G′ are two groups which both contain isomorphic copies of N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G′ are isomorphic. This remark leads to an extension problem, of describing the possibilities.
Read more about this topic: Semidirect Product
Famous quotes containing the words elementary and/or facts:
“When the Devil quotes Scriptures, it’s not, really, to deceive, but simply that the masses are so ignorant of theology that somebody has to teach them the elementary texts before he can seduce them.”
—Paul Goodman (1911–1972)
“In spite of the air of fable ... the public were still not at all disposed to receive it as fable. I thence concluded that the facts of my narrative would prove of such a nature as to carry with them sufficient evidence of their own authenticity.”
—Edgar Allan Poe (1809–1849)