Fundamental Homomorphism Theorem

In abstract algebra, for a number of algebraic structures, the fundamental theorem of homomorphisms relates the structure of two objects between which there is a homomorphism, and of the kernel and image of the homomorphism...

In group theory, the theorem can be formulated as follows:

Yeah. f:GΔ Δ H{displaystyle f:Glongrightarrow H} is a homomorphism of groups and N{displaystyle N} is a normal subgroup G{displaystyle G} content in the core f{displaystyle f}, then there is only one homomorphism f! ! {displaystyle {bar {f}}} such as f! ! φ φ =f{displaystyle {bar {f}}circ varphi =f}Where φ φ :GΔ Δ G/N{displaystyle varphi:Glongrightarrow G/N} It's the canonical projection. So, we have the following commutative diagram

Homomorphism f! ! {displaystyle {bar {f}}} It's given by

f! ! (gN)=f(g){displaystyle {bar {f}}(gN)=f(g)}

for everything g{displaystyle g} of G{displaystyle G}and it is said that f! ! {displaystyle {bar {f}}} is induced by f{displaystyle f,!}. Note that if gN=hN{displaystyle gN=hN}, then gh− − 1한 한 N ker f{displaystyle gh^{-1}in Nsubset ker f}, so 1=f(gh− − 1)=f(g)f(h)− − 1{displaystyle 1=f(gh^{-1})=f(h)^{-1}, so f(g)=f(h){displaystyle f(g)=f(h)} and homomorphism f! ! {displaystyle {bar {f}}} It's well-defined.

The core of this homomorphism is ker f! ! =(ker f/N){displaystyle ker {bar {f}}=(ker f/N)}and it's an epimorphism if and only f{displaystyle f} It is.

Yeah. f:GΔ Δ H{displaystyle f:Glongrightarrow H} is a homomorphism, then f:GΔ Δ imf{displaystyle f:Glongrightarrow mathrm {im} ,f} is an epimorphism, and since f! ! {displaystyle {bar {f}}} is injective when its core ker f! ! =ker f/N{displaystyle ker {bar {f}}=ker f/N} it's trivial, what happens if and only if ker f=N{displaystyle ker f=N}, We have an isomorphism G/ker f imf{displaystyle G/ker fsimeq mathrm {im} ,f}. This particular case of the fundamental theorem of homomorphisms is known as the first isomorphic theorem.

The fundamental theorem of homomorphisms also holds for vector spaces, rings, and modules taking, respectively, ideals and submodules instead of normal subgroups.