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单词 DecomposableHomomorphismsAndFullFamiliesOfGroups
释义

decomposable homomorphisms and full families of groups


Let {Gi}iI,{Hi}iI be two families of groups (indexed with the same set I).

Definition. We will say that a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath

f:iIGiiIHi

is decomposableMathworldPlanetmathPlanetmathPlanetmath if there exists a family of homomorphisms {fi:GiHi}iI such that

f=iIfi.

Remarks. For each jI and giIGi we will say that gGj if g(i)=0 for any ij. One can easily show that any homomorphism

f:iIGiiIHi

is decomposable if and only if for any jI and any giIGi such that gGj we have f(g)Hj. This implies that if f is an isomorphismMathworldPlanetmathPlanetmathPlanetmath and f is decomposable, then each homomorphism in decomposition is an isomorphism and

(iIfi)-1=iIfi-1.

Also it is worthy to note that compositionMathworldPlanetmathPlanetmath of two decomposable homomorphisms is also decomposable and

(iIfi)(iIgi)=iIfigi.

Definition. We will say that family of groups {Gi}iI is full if each homomorphism

f:iIGiiIGi

is decomposable.

Remark. It is easy to see that if {Gi}iI is a full family of groups and I0I, then {Gi}iI0 is also a full family of groups.

Example. Let 𝒫={p|p is prime}. Then {p}p𝒫 is full. Indeed, let

f:p𝒫pp𝒫p

be a group homomorphism. Then, for any q𝒫 and ap𝒫p such that aq we have that |a| divides q and thus |f(a)| divides q, so it is easy to see that f(a)q. Therefore (due to first remark) f is decomposable.

Counterexample. Let G1,G2 be two copies of . Then {G1,G2} is not full. Indeed, let

f:

be a group homomorphism defined by

f(x,y)=(0,x+y).

Now assume that f=f1f2. Then we have:

(0,1)=f(1,0)=(f1(1),f2(0))

and so f2(0)=1. ContradictionMathworldPlanetmathPlanetmath, since group homomorphisms preserve neutral elements.

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更新时间:2025/5/4 16:54:42