# Subgroups between $S_n$ and $S_{n+1}$

Lets look at $S_n$ as subgroup of $S_{n+1}$. How many subgroups $H$, $S_{n} \subseteq H \subseteq S_{n+1}$ there are ?

-

1
What can you say about an element of $H$ which is not in $S_n$? – Qiaochu Yuan Jul 19 '12 at 15:33
Stabilizers in primitive (hence 2-transitive) groups are maximal. – Steve D Jul 19 '12 at 18:25

## 1 Answer

None.

Let $S_n<H<S_{n+1}$ and suppose that $H$ contains a cycle $c$ involving $n+1$, say $$c=(a,\cdots,b,n+1).$$ Then by composing to the left with a suitable permutation $\sigma\in S_n<H$ such that $\sigma(a)=b$ we have $$\sigma c=\sigma^\prime (b,n+1)$$ where $\sigma^\prime(n+1)=(n+1)$, i.e. $\sigma^\prime\in H$. Thus the transposition $(b,n+1)$ is in $H$. Of course we may assume that $b=1$ and so all transpositions $(1,2)$, $(1,3)$, ..., $(1,n+1)$ are in $H$. These transpositions are known to generate $S_{n+1}$.

-
Sorry I used a non-standard notation for cycles (the commas aren't usually there) but I couldn't find a way to manage spaces in this TeX environment. – Andrea Mori Jul 19 '12 at 15:39

4
If by none you mean two... (the inclusions are not necessarily proper on either end!) – Qiaochu Yuan Jul 19 '12 at 15:40

1
I often put in ~'s for cycle notation. e.g. `$(1~2~3)$`=$(1~2~3)$ – anon Jul 19 '12 at 15:40

1
@QiaochuYuan: of course you are right. The "none" incipit was chosen for dramatic effects :) I guess that I should say that I was impliciting assuming that $H$ is not one of the trivial possibilities. – Andrea Mori Jul 19 '12 at 15:45
@anon: thanks for the tip. – Andrea Mori Jul 19 '12 at 15:46
{"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": 26, "mathjax_display_tex": 2, "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": 0.9465872049331665, "perplexity_flag": "middle"}