mathgirl1
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Let G be a group of order pm where p is a prime and p > m. Suppose H is a subgroup of order p. Show that H is normal in G.
There is a very similar problem
Let |G| = p^nm where p is a prime and n \ge 1, p > m. Show that the Sylow p-subgroup of G is normal in G.
Proof:
Let n_p be the number of Sylow p-subgroups of G. By the 3-d Sylow Subgroup Theorem we know that n_p \mid m and that n_p \equiv 1 (mod p). Since m < p and n_p \mid m, it follows that 1 \le n_p \le m < p. Since we also know that n_p \equiv 1 (mod p), it follows that n_p=1. Let P be a Sylow p-subgroup of G. Since for every g \in G, g^{-1}Pg is also a Sylow p-subgroup of G and since n_p=1, it follows that for every g \in G that g^{-1}Pg=P. Hence P is normal in G, as claimed.
So this is the same problem that I need to solve with n=1. Is there a way to solve this without Sylow Theorems or Sylow subgroups? We barely covered that so not sure we can use it to prove this problem.
Also, why is g^{-1}Pg also a Sylow p-subgroup of G since P is?
Any help is appreciated. I would like to solve this problem without Sylow theorems but not sure where to start cause I am already stuck in this thinking.
Thanks!
There is a very similar problem
Let |G| = p^nm where p is a prime and n \ge 1, p > m. Show that the Sylow p-subgroup of G is normal in G.
Proof:
Let n_p be the number of Sylow p-subgroups of G. By the 3-d Sylow Subgroup Theorem we know that n_p \mid m and that n_p \equiv 1 (mod p). Since m < p and n_p \mid m, it follows that 1 \le n_p \le m < p. Since we also know that n_p \equiv 1 (mod p), it follows that n_p=1. Let P be a Sylow p-subgroup of G. Since for every g \in G, g^{-1}Pg is also a Sylow p-subgroup of G and since n_p=1, it follows that for every g \in G that g^{-1}Pg=P. Hence P is normal in G, as claimed.
So this is the same problem that I need to solve with n=1. Is there a way to solve this without Sylow Theorems or Sylow subgroups? We barely covered that so not sure we can use it to prove this problem.
Also, why is g^{-1}Pg also a Sylow p-subgroup of G since P is?
Any help is appreciated. I would like to solve this problem without Sylow theorems but not sure where to start cause I am already stuck in this thinking.
Thanks!