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September 30th, 2017, 03:22 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0  $H$ is relatively prime to $[G:H]$
Ineed help to solve the following problem: Let $G$ be a finite group and $H<G$. suppose that $C_G(x)\subset H$ for all $x\in H\backslash\left\{e\right\}.$ Show that $H$ is relatively prime to $[G:H]$. Thanks in advance. Last edited by mona123; September 30th, 2017 at 03:34 AM. 
October 3rd, 2017, 10:23 AM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 
For p a prime, you just need to use 2 elementary properties of a finite p group P. 1. If Q is a proper subgroup of P, $Q\subset N_P(Q)$ 2. If $<1>\neq N$ is a normal subgroup of P, $N\cap Z(P)\neq <1>$ 

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$gh$, $h$, prime 
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