TITLE: Simple groups of order 300
QUESTION [5 upvotes]: I am aware of No simple group of order $300$.
It is said that "there would be 6 Sylow 5-groups, one of which will have an index of 6", but why does one have index 6? If we write $|G|=p^k m$ where $p\nmid m$, then by Lagrange's theorem any $p$-Sylowgroup has index $m$, which in this case is 12?

REPLY [6 votes]: What you said is said in that link is a serious blunder: as all the Sylow $\,5-$subgroups obviously have the same order then they all have the same index. 
What is true is that the number of different Sylow $\,5-$subgroups equals the index of the normalizer of any of them, so we have a subgroup $\,N\le G\,$ with $\,[G:N]=6\;$. 
This already leads to a contradiction if we suppose $\,G\;$  is simple, as then the regular action of $\,G\,$ on $\,N\,$ gives us an injective homomorphism $\,G\to S_6\,$...but this is impossible (why?) .