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I know that the order of an element divides the order of the group , but what about a subgroup ?

Say for instance you know the elements in a sylow p- subgroup have order p ( p a prime ) then does that mean that p must divide the order of this subgroup aswell or just the larger group $G=p^{\alpha }m$

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    $\begingroup$ A subgroup is simply a group on its own (with operations inherited from the ambient group) so all the properties that exists between a group and it's elements would hold true with a subgroup and it's elements. $\endgroup$ – Anurag A Nov 6 '18 at 19:57
  • $\begingroup$ Yes. A subgroup is a group itself, so as long as the element resides in that subgroup its order will divide the order of the subgroup. More broadly, any theorem that applies to groups necessarily applies to subgroups, as subgroups are themselves groups. $\endgroup$ – David Reed Nov 6 '18 at 19:57
  • $\begingroup$ Ah. you see I was considering a group of order 120 and then wanted to find its 5 sylow subgroups obviously 120 factorises as 5.24. which according to sylows theorem implies that the 5 sylow subgroups have order 5. then i thought i had come to a contradiction because the number of sylow 5 subgroups is 1 or 6. I know see the source of my confusion was that $n_5$ is just the number of subgroups not the order of the subgroup. and infact by my first assumption the order of the group must be 5 $\endgroup$ – excalibirr Nov 6 '18 at 20:02
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Yes. If $g\in H$ and $H$ is a subgroup of $G$, then the order of $g$ as an element of $H$ is equal to the order of $g$ as an element of $G$. And, since the order of an element divides the order of the group, the order of $g$ divides the order of $H$.

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