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September 30th, 2017, 04:22 AM   #1
Joined: Jan 2015
From: usa

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$|H|$ 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 04:34 AM.
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October 3rd, 2017, 11:23 AM   #2
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From: Athens, OH

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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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