1

I'm studying group theory for a new exam that I will have to take in 3-4 months. I was looking for some exercises online and I found my Professor's page, where I found the following:

Let $G$ be a finite group and let $H$ and $K$ be normal subgroups of $G$ where $G = HK$. Consider $P$ a p-Sylow subgroup of $G$. Show that $P = (P \cap H)(P \cap K)$.

One of the inclusion is trivial, but what about the other one? I've tried to use properties of powers as suggested by a friend, but I'm unable to conclude anything. I'm stuck :( I'm familiar with the meaning of the objects written above, however, I do not have yet the ability to show the result on my own. If someone of you could explain to me the logic behind this (kind of) proof(s), I'd be more than excited to read it!

Thank you so much!

eallaipigreco
  • 11
  • 1
    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Dec 15 '22 at 22:18
  • 1
    Thank you for your comment! I totally understand what you say. I've been trying to solve this exercise for several hours but I don't seem to succeed :) Unfortunately, I'm not an English speaker and I'm not confident with writing maths in English, this is why I'm struggling with adding more details. Sorry again! – eallaipigreco Dec 15 '22 at 22:24
  • You're welcome! Your English is fine. That's alright; instead, try answering the following questions (in an [edit]): What are you studying? What text is this drawn from, if any? If not, how did the question arise? What kind of approaches (to similar problems) are you familiar with? What kind of answer are you looking for? Basic approach, hint, explanation, something else? Is this question something you think you should be able to answer? Why or why not? – Shaun Dec 15 '22 at 22:30
  • Hang on: I just saw the duplicate. Never mind! – Shaun Dec 15 '22 at 22:31
  • I edited it, hope it's better. I saw the duplicate but I don't see how these things are related... In the example above, G is not a direct product (in fact, the intersection between $H$ and $G$ is not necessarily trivial). – eallaipigreco Dec 15 '22 at 22:40
  • 1
    If you consider $G/(H\cap K)$, you get the duplicate problem. – Arturo Magidin Dec 15 '22 at 22:41
  • 1
    Moreover, the accepted answer solves an even more general problem, not assuming that $H$ and $K$ are normal. It shows that if $G=HK$ then there exists a Sylow $p$-subgroup $P$ of $G$ such that $P=(H\cap P)(K\cap P)$ with the additional condition that $H\cap P$ is a Sylow $p$-subgroup of $H$ and $K\cap P$ is a Sylow $p$-subgroup of $G$. If you further assume $H$ and $K$ are normal, and $Q$ is any Sylow $p$-subgroup of $G$, then conjugating $P$ to get $Q$ gives you the result for all Sylow $p$-subgroup of $G$. – Arturo Magidin Dec 15 '22 at 22:45

0 Answers0