# fourth isomorphism theorem

###### Theorem 1 (The Fourth Isomorphism Theorem)

Let $G$ be a group and $N\unlhd G$. There is a bijection between $G(N)$, the set of subgroups of $G$ containing $N$, and the set of subgroups of $G/N$ defined by $A\rightarrow A/N$. Moreover, for any two subgroups $A,B$ in $G(N)$, we have

1. 1.

$A\leq B$ if and only if $A/N\leq B/N$,

2. 2.

$A\leq B$ implies $|B:A|=|B/N:A/N|$,

3. 3.

$\langle{A,B}\rangle/N=\langle{A/N,B/N}\rangle$,

4. 4.

$(A\cap B)/N=(A/N)\cap(B/N)$, and

5. 5.

$A\unlhd G$ if and only if $(A/N)\unlhd(G/N)$.

Title fourth isomorphism theorem FourthIsomorphismTheorem 2013-03-22 14:00:52 2013-03-22 14:00:52 bwebste (988) bwebste (988) 16 bwebste (988) Theorem msc 20A05 correspondence theorem lattice isomorphism theorem