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Let $G$ be a group of order $21$ contains an element $a$ of order $7$ . Prove that $A$=($a$) ,the subgroup generated by $a$ , is normal in $G$ .

I'm more concerned with how I can derive the prove of this question

lio
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  • It's different question, If you can not answer it, don't only judge the question. I need help from someone else who can answer it. By the way this question is in section before learning sylow theorem in my book @dietrichburde – lio Apr 05 '17 at 14:38
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    Why is it different? There it is shown that the $7$-Sylow subgroup is normal, and your subgroup $A$ is exactly the Sylow $7$-subgroup. – Dietrich Burde Apr 05 '17 at 14:44
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    It also follows from this duplicate. Since $p=3$ is the smallest prime divisor of $21$, and the index of the subgroup $A$ is $3$, the subgroup $A$ must be normal - without Sylow. – Dietrich Burde Apr 05 '17 at 14:46
  • By the way this question is in section before learning sylow theorem in my book @DietrichBurde . means that this problem can be solved without sylow actually – lio Apr 05 '17 at 14:49
  • Then use the duplicate with index $p=3$, which is without Sylow. – Dietrich Burde Apr 05 '17 at 14:51
  • We can't read your mind, lio. When you ask a question, you should be as specific as possible: sharing things like, what you have covered thus far in the class, or being involved as least to providing the definition of "normal subgroup" that you've working with. Folks can then tailor answers and suggested posts that won't rely on things you haven't covered yet. – amWhy Apr 05 '17 at 14:54
  • that's not duplicate. You can use that question (with index p=3) to answer my question but their not same or duplicate I think – lio Apr 05 '17 at 14:56
  • lio: you need to understand that the same question can be asked in different ways, and also answered in different ways. It's up to you to tell us from the get-go what tools you have been given, are available for you to use. – amWhy Apr 05 '17 at 15:03
  • lio I've already addressed how you can improve your post, by identifying definitions you've learned (or have been presented in your text/course thus far. All you've done is further claim your question is not a duplicate. You've haven't chosen to do even that. – amWhy Apr 05 '17 at 15:06
  • @amWhy yes, the same question can be asked in different ways. But this different. If you lazy to answer it, You can give me the other question that maybe help me to answer this question but don't judge the question is same. if I knew about it was there, I may not ask this again. – lio Apr 05 '17 at 15:12
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    Before accusing any potential answerer of being lazy by not answering your question, look in the mirror, and you'll see the real user who's being lazy. – amWhy Apr 05 '17 at 15:17
  • Actually you did not answer it, right? just claim that they are the same thing. They are similar but not same. you can look up whether there is your answer or not before ordering me to looking my self in the mirror – lio Apr 05 '17 at 15:21
  • What is true is that @DietrichBurde has gone to the trouble of finding duplicate questions, with answers addressing your question. It is also true that I'm spent time searching for related posts. The thing is, that's your job, not ours. You are the one who is responsible for searching the site before asking a question. It is your responsibility to provide sufficient context as to what you've learned are or now learning, any any other information that would help a potential answerer to address an answer at your level. – amWhy Apr 05 '17 at 15:35
  • And please remember no one on this site, askers, answerers and even the site mods, is paid to work here. We all volunteer our time here. So no one is entitled to get an answer. Users are free to ask, and others to answer, or search,, or none of the above. But there is not guarantee that a question will be answered, especially when the question is of poor quality. – amWhy Apr 05 '17 at 15:38

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