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.
$A\\leq B$ if and only if $A/N\\leq B/N$ ,
2.
$A\\leq B$ implies $|B:A|=|B/N:A/N|$ ,
3.
$\\langle{A,B}\\rangle/N=\\langle{A/N,B/N}\\rangle$ ,
4.
$(A\\cap B)/N=(A/N)\\cap(B/N)$ , and
5.
$A\\unlhd G$ if and only if $(A/N)\\unlhd(G/N)$ .

单词	FourthIsomorphismTheorem
释义	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. $A\\leq B$ if and only if $A/N\\leq B/N$ , 2. $A\\leq B$ implies $\|B:A\|=\|B/N:A/N\|$ , 3. $\\langle{A,B}\\rangle/N=\\langle{A/N,B/N}\\rangle$ , 4. $(A\\cap B)/N=(A/N)\\cap(B/N)$ , and 5. $A\\unlhd G$ if and only if $(A/N)\\unlhd(G/N)$ .
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