fourth isomorphism theorem

Theorem 1 (The Fourth Isomorphism Theorem)

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

1. $A\le B$ if and only if $A/N\le B/N$,

2. $A\le B$ implies $|B:A|=|B/N:A/N|$,

3. $⟨A,B⟩/N=⟨A/N,B/N⟩$,

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

5. $A⊴G$ if and only if $(A/N)⊴(G/N)$.

Title	fourth isomorphism theorem
Canonical name	FourthIsomorphismTheorem
Date of creation	2013-03-22 14:00:52
Last modified on	2013-03-22 14:00:52
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	16
Author	bwebste (988)
Entry type	Theorem
Classification	msc 20A05
Synonym	correspondence theorem
Synonym	lattice isomorphism theorem