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Do Sylow's theorems tell us that there exists at least one group of order $p^i , i=\{1,2, ..., n\}$ if $|G|=p^na$ ?

The textbook that I have doesn't say anything about that, is this trivial am I missing something ?

For example let $|G|=5^9\cdot 3$ does that mean that $G$ has subgroups with order $5,5^2,5^3,...,5^9$ ?

Arturo Magidin
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  • Which textbook are you referring to? – Shaun Oct 27 '21 at 19:27
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    There are multiple ways of stating Sylow's Theorems. Some of them include this as part of their statement. Other ways of stating it do not, and you must deduce it from other results, such as the existence of Sylow $p$-subgroups and the class formula. So you need to say exactly how the theorems are phrased, and what other results you may or may not know related to this. – Arturo Magidin Oct 27 '21 at 19:28
  • @Shaun it's a Greek textbook, you probably don't know it. – 領域展開 Oct 27 '21 at 19:30
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    In my experience, existence is pretty commonly included, but I suppose that need not be universally true. (Would seem strange not to.) – Randall Oct 27 '21 at 19:31
  • @Randall ok, it's true then, I will try to prove it if so. – 領域展開 Oct 27 '21 at 19:32
  • Still: it might be of use to someone. I didn't ask for my benefit. Please share the details. – Shaun Oct 27 '21 at 19:33
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    You didn't ask, but to me the most readable treatment of all this is in Fraleigh's book. I was confused about this for far too long before I encountered Fraleigh's treatment (which is not unique to Fraleigh). – Randall Oct 27 '21 at 19:38
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    @Shaun the book is called "an introduction to algebra" it's by the EKPA university, I don't think it's even translated in English. – 領域展開 Oct 27 '21 at 19:40
  • Thank you. Some users here know Greek. – Shaun Oct 27 '21 at 19:45
  • https://math.stackexchange.com/questions/549635/a-p-group-of-order-pn-has-a-normal-subgroup-of-order-pk-for-each-0-le-k – Aaron Oct 27 '21 at 20:15

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