# Subgroup that contains all Sylow $p$-subgroups

Let $P$ be a Sylow $p$-subgroup of a finite group $G$, for some prime $p$. Prove that if $H$ is a subgroup of $G$ that contains all Sylow $p$-subgroups of $G$, then $G = HN_G(P)$.

Here's what I have so far:

I know that the normalizer of $P$ in $G$ is $$N_G(P) = \left\{g \in G \,\mid\, gAg^{-1} = A\right\}$$

where $$gAg^{-1} = \left\{ gag^{-1}\,\mid\, a \in A \right\}$$

and that $HN_G(P)$ refers to the product of subgroups.

If the order of $G$ is $p^n$ for some $n \in \mathbb{N}$, then the multiplicity of $p$ is $n$. This means $G$ is a Sylow $p$-subgroup.

If we choose $P = G$, then $H$ must contain $P$, so $H = G$. $N_G(G) = G$, so $$HN_G(P) = GG = G$$

This isn't remotely a proof, only a case in which what I'm trying to prove is true. A hint would be great. Thanks in advance.

• Why do you think $\;HN_G(P)\;$ is a direct product? It is a product of groups, not necessarily direct. – DonAntonio Dec 14 '13 at 16:12
• Thanks - should be a group product. I'll edit the question. – Zubin Mukerjee Dec 14 '13 at 16:14

You could "copy" the proof of Frattini's Argument: let $\;g\in G\;$ and let $\;P\;$ be some Sylow subgroup of $\;G\;$ , so $\;P\le H\;$ . But also $\;P^g:=g^{-1}Pg\;$ is a Sylow subgroup and thus also $\;P^g\le H$ .
Note that all the Sylow subgroups of $\;G\;$ are also Sylow subgps. of $\;H\;$ , so by Sylow's Theorems
$$\exists\;h\in H\;\;s.t.\;\;P^g=P^h\implies P^{gh^{-1}}=P\implies gh^{-1}\in N_G(P)$$
{"ft_lang_label":"__label__en","ft_lang_prob":0.8960402,"math_prob":0.9999776,"size":787,"snap":"2019-26-2019-30","text_gpt3_token_len":296,"char_repetition_ratio":0.11749681,"word_repetition_ratio":0.0,"special_character_ratio":0.386277,"punctuation_ratio":0.1160221,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000009,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-20T22:08:37Z\",\"WARC-Record-ID\":\"<urn:uuid:d2a02822-d4be-46c3-bed4-9ce865b90afa>\",\"Content-Length\":\"141024\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:31dabab8-c178-4b7d-9735-5fef20a10283>\",\"WARC-Concurrent-To\":\"<urn:uuid:b2ce4d8e-3d18-412d-944a-d68509f1e5ea>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/606660/subgroup-that-contains-all-sylow-p-subgroups\",\"WARC-Payload-Digest\":\"sha1:DT6ODEV7XO7NJY4FODUZLIZVXETAGXKF\",\"WARC-Block-Digest\":\"sha1:ABTOHEKZMH4QKOKVFVFEG2MNZOQYHDGE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195526714.15_warc_CC-MAIN-20190720214645-20190721000645-00248.warc.gz\"}"}