Group Theoretic Version
Given two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K (where G/K is a quotient group). If K is a subset of ker(f) then there exists a unique homomorphism h:G/K→H such that f = h φ.
The situation is described by the following commutative diagram:
By setting K = ker(f) we immediately get the first isomorphism theorem.
Read more about this topic: Fundamental Theorem On Homomorphisms
Famous quotes containing the words group and/or version:
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)
“It is never the thing but the version of the thing:
The fragrance of the woman not her self,
Her self in her manner not the solid block,
The day in its color not perpending time,
Time in its weather, our most sovereign lord,
The weather in words and words in sounds of sound.”
—Wallace Stevens (1879–1955)