Isomorphism theorem

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In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms.

Groups

First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of quotient groups (also called factor groups). All three involve "modding out" by a normal subgroup.

First isomorphism theorem

If G and H are groups and f is a homomorphism from G to H, then the kernel K of f is a normal subgroup of G, and the quotient group G/K is isomorphic to the image of f.

If

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G, H \mbox{ are groups}\;}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: G \to H, f \mbox{ is a homomorphism}\;}

then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ker}(f) \triangleleft G }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/\operatorname{Ker}(f) \cong \operatorname{Im}(f)}

Second isomorphism theorem

Let H and K be subgroups of the group G, and assume H is a subgroup of the normalizer of K. Then the join HK of H and K is a subgroup of G, K is a normal subgroup of HK, H ∩K is a normal subgroup of H, and HK/K is isomorphic to H/(H ∩K).

If

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,K \mbox{ are subgroups of group } G \,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \mbox{ is a subgroup of } \operatorname{N_G}(K)}

then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HK \mbox{ is a subgroup of } G \,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \triangleleft HK }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \cap K \triangleleft H}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HK/K \cong H/(H \cap K)}

Third isomorphism theorem

If M and N are normal subgroups of G such that M is contained in N, then M is a normal subgroup of N, N/M is a normal subgroup of G/M, and (G/M)/(N/M) is isomorphic to G/N.

If

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M,N \triangleleft G}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \subseteq N}

then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \triangleleft N}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N/M \triangleleft G/M}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G/M)/(N/M) \cong G/N}

Rings and modules

The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field). One has to replace the term "group" by "R-module", "subgroup" and "normal subgroup" by "submodule", and "factor group" by "factor module".

The isomorphism theorems are also valid for rings, ring homomorphisms and ideals. One has to replace the term "group" by "ring", "subgroup" and "normal subgroup" by "ideal", and "factor group" by "factor ring".

The notation for the join in both these cases is "H + K" instead of "HK".

We also need to mention the isomorphism theorems for topological vector spaces, Banach algebras etc.

General

To generalise this to universal algebra, normal subgroups need to be undermined by congruences.

de:Isomorphiesatz he:משפטי האיזומורפיזם (אלגברה)