# Correspondence between normal subgroups of a group and normal subgroups of a quotient groups

Suppose $G$ be a group and $N$ be a proper normal subgroup such that it is not contained in any other proper normal subgroup. Does that mean $G/N$ is simple?

Is there one-one correspondence between the normal subgroups of $G/N$ and normal subgroups of $G$ containing $N$?

-

5
Yes. The correspondence between the subgroups of $G/N$ and those of $G$ which contain $N$ maps normal subgroups to normal subgroups. – Mariano Suárez-Alvarez♦ Sep 26 '11 at 17:58

## 1 Answer

The lattice isomorphism between subgroups of $G$ that contain $N$ and subgroups of $G/N$ is not only inclusion-preserving, but also normality-preserving.

If $N\lt H\lt G$ and $H\triangleleft G$, then for all $h\in H$ and $g\in G$ we have $ghg^{-1}\in H$. Then for all $gN\in G/N$ and all $hN\in H/N$ we have $(gN)(hN)(gN)^{-1} = gNhNg^{-1}N = ghg^{-1}N\in HN$, so $gNhN(gN)^{-1}\in H/N$. Hence, the image of a normal subgroup under the correspondence is normal in the quotient.

Conversely, if $H/N$ is normal in $G/N$, and $g\in G$, $h\in H$, then we know that $ghg^{-1}N\in HN$, so there exists $h'\in H$ and $n\in N$ such that $ghg^{-1}=h'n$. But $N\subseteq H$, so $h'n\in H$. Hence, $ghg^{-1}\in H$ for all $h\in G$, $g\in G$, so $H\triangleleft G$; showing that if the image of the subgruop is normal, then the original subgroup is normal as well.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 35, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.911108136177063, "perplexity_flag": "head"}