If a group G has order [itex] p^n [/itex], show that any subgroup of order [itex] p^{n-1} [/itex] is normal in G.(adsbygoogle = window.adsbygoogle || []).push({});

i have no idea how to start this. i know that to show that a subgroup N is normal in G, i need that [itex] gNg^{-1} = N [/itex]. so i start with any subgroup N of order [itex] p^{n-1} [/itex] but i have no idea how to continue.

this problem appears in the section before the sylow theorems are introduced so i can't use them. i know that for p-groups, the center is nontrivial and has prime power order. also in this same section, Cauchy's theorem was introduced which says if p divides the order of a group then that group has an element of order p. these concepts were introduced fairly recently to me so this may be why i am having trouble.

can someone give me a hint or 2 to continue? thanks

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# Homework Help: If a group G has order p^n, show any subgroup of order p^n-1 is normal

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