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

• Vector spaces over a body
• or Abelian groups
• or more generally modules over a ring
• or objects of an Abelian category in general .

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 .

literature

• Siegfried Bosch : Algebra. 8th edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-642-39566-6 , chapter 1.2.
• Christian Karpfinger, Kurt Meyberg: Algebra. 3. Edition. Springer, Berlin / Heidelberg 2013, ISBN 9783827430113 , chapter 4.6.

Web links

matheplanet.com: Peer pressure IV - Detailed explanations and proofs of the isomorphism theorems