Isomorphism theorem

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

$\displaystyle G, H \mbox{ are groups}\;$
$\displaystyle f: G \to H, f \mbox{ is a homomorphism}\;$

then

$\displaystyle \operatorname{Ker}(f) \triangleleft G$
$\displaystyle G/\operatorname{Ker}(f) \cong \operatorname{Im}(f)$

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

$\displaystyle H,K \mbox{ are subgroups of group } G \,$
$\displaystyle H \mbox{ is a subgroup of } \operatorname{N_G}(K)$

then

$\displaystyle HK \mbox{ is a subgroup of } G \,$
$\displaystyle K \triangleleft HK$
$\displaystyle H \cap K \triangleleft H$
$\displaystyle HK/K \cong H/(H \cap K)$

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

$\displaystyle M,N \triangleleft G$
$\displaystyle M \subseteq N$

then

$\displaystyle M \triangleleft N$
$\displaystyle N/M \triangleleft G/M$
$\displaystyle (G/M)/(N/M) \cong G/N$

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.