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I am working on the same problem here: A simple group of order $168$ doesn't have subgroups of order $14$

I have prove most of the requirements, but I don't know what $|N_G(P)|$ means, I haven't done it in my course yet. Can someone give the definition, a simple google could do it, thanks.

Shaun
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  • Cf. normalizer of the subset $P$ in the group $G$ – J. W. Tanner Oct 26 '21 at 17:34
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    The cardinality (number of elements in) $N_G(P) = {g\in G\mid gxg^{-1}\in P\text{ for all }x\in P}$. Both the bar notation and $N_G(P)$ are pretty standard; it would be hard to work on questions about simple groups if you've never seen the normalizer! – Arturo Magidin Oct 26 '21 at 17:38

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For any $S\subseteq G$, we denote by $\lvert S\rvert $ the cardinality of the set $S$.

The notation

$$N_G(P)=\{ g\in G\mid gP=Pg\}$$

is the normaliser of $P\subseteq G$ in $G$, where

$$gP=\{ gp\in G\mid p\in P\}$$

and

$$Pg=\{ pg\in G\mid p\in P\}.$$

Shaun
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