THEOREM OF THE DAY: Sylow’s Theorems

Theorem

Let $G$ be a finite group of order pnm where $p$ is a prime not dividing $m$. 

Then 

1. $G$ has subgroups of order $p^n$. Any subgroup of this order, termed a $p$-Sylow (pronounced, for anglophones, roughly as in ‘Seal of approval’) subgroup, is conjugate to any other; 

2. any subgroup of $G$ of p-power order is contained in a $p$-Sylow subgroup; 

3. the number of $p$-Sylow subgroups divides m, and is $1$ more than a multiple of $p$. 

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