# Isomorphism Theorem

The isomorphism theorems are two mathematical theorems that make statements about groups . They can also be transferred to more complex algebraic structures and are therefore an important result of universal algebra . The isomorphism theorems are a direct consequence of the homomorphism theorem of the corresponding algebraic structure .

Sometimes the homomorphism set is called the first isomorphism set. The sentences given below are then called the second or third isomorphism sentence .

## Group theory

### First isomorphism theorem

Let there be a group, a normal subgroup in and a subgroup of . Then the complex product is also a subgroup of , is a normal part of and the group is normal part of in . The following applies: ${\ displaystyle G}$${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle H}$${\ displaystyle G}$ ${\ displaystyle HN: = \ {hn \ mid h \ in H, n \ in N \}}$${\ displaystyle G}$${\ displaystyle N}$${\ displaystyle HN}$${\ displaystyle H \ cap N}$${\ displaystyle H}$

${\ displaystyle H / (H \ cap N) \ cong HN / N.}$

The isomorphism of groups denotes . ${\ displaystyle \ cong}$

The isomorphism that is usually meant is called canonical isomorphism. According to the homomorphism theorem, it is derived from the surjective mapping

${\ displaystyle f \ colon H \ to HN / N, \ quad h \ mapsto hN,}$

induced, because it obviously holds

${\ displaystyle \ mathrm {kern} \ left (f \ right) = \ left \ {a \ in H \ mid aN = N \ right \} = \ left \ {a \ in H \ mid a \ in N \ right \} = H \ cap N}$.

From the first isomorphism theorem, as a special case, one receives the clear statement that one can "expand" with exactly when . ${\ displaystyle N}$${\ displaystyle H \ cap N = \ {0 \}}$

### Second isomorphism theorem

Let there be a group, a normal subgroup in and a subgroup of which is normal subgroup in . Then: ${\ displaystyle G}$${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle N}$${\ displaystyle H}$${\ displaystyle G}$

• ${\ displaystyle (G / N) / (H / N) \ cong G / H.}$

In this case canonical isomorphisms can be given in both directions, induced by on the one hand

${\ displaystyle G / N \ to G / H, \ quad gN \ mapsto gH,}$

on the other hand through

${\ displaystyle G \ to (G / N) / (H / N), \ quad g \ mapsto gN (H / N).}$

To put it clearly, the second isomorphism statement says that one can "shorten". ${\ displaystyle N}$

## Rings

The isomorphism theorems also apply to rings in an adapted form:

### First isomorphism theorem

Let it be a ring, an ideal of and a sub-ring of . Then the sum is a subring of and the cut is an ideal of . The following applies: ${\ displaystyle R}$${\ displaystyle {\ mathfrak {a}}}$${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle R}$${\ displaystyle S + {\ mathfrak {a}} = \ {s + a | s \ in S, a \ in {\ mathfrak {a}} \}}$${\ displaystyle R}$${\ displaystyle S \ cap {\ mathfrak {a}}}$${\ displaystyle S}$

${\ displaystyle S / (S \ cap {\ mathfrak {a}}) \ cong (S + {\ mathfrak {a}}) / {\ mathfrak {a}}}$

The isomorphism of rings denotes . ${\ displaystyle \ cong}$

### Second isomorphism theorem

Let it be a ring, two ideals of . Then is an ideal of . The following applies: ${\ displaystyle R}$${\ displaystyle {\ mathfrak {b}} \ subseteq {\ mathfrak {a}}}$${\ displaystyle R}$${\ displaystyle {\ mathfrak {a}} / {\ mathfrak {b}} = \ {a + {\ mathfrak {b}} | a \ in {\ mathfrak {a}} \}}$${\ displaystyle R / {\ mathfrak {b}}}$

${\ displaystyle (R / {\ mathfrak {b}}) / ({\ mathfrak {a}} / {\ mathfrak {b}}) \ cong R / {\ mathfrak {a}}}$

## Vector spaces, Abelian groups, or objects of any Abelian category

Be there ${\ displaystyle M, N \ subseteq Q \ subseteq P}$

Then:

• ${\ displaystyle M / (M \ cap N) \ cong (M + N) / N}$
• ${\ displaystyle (P / N) / (Q / N) \ cong P / Q}$

Here, too, the symbol stands for the isomorphism of the corresponding algebraic structures or objects in the respective category. ${\ displaystyle \ cong}$

The canonical isomorphisms are clearly determined by the fact that they are compatible with the two canonical arrows of or . ${\ displaystyle M}$${\ displaystyle P}$

The serpent lemma provides a far-reaching generalization of the isomorphism theorems .