Monthly Archives: March 2017

A Clever Proof of a Common Fact

Cf. means confer…

Abstract Nonsense

Point of Post: In this post we give a new proof that if $latex G$ is a finite group $latex H$ is a subgroup $latex G$ whose index is the smallest prime dividing $latex |G|$ then that subgroup is normal.

$latex text{ }$

Motivation

It is a commonly used theorem in finite group theory that if $latex G$ is a finite group and $latex Hleqslant G$ such that $latex left[G:Hright]=p$ is the smallest prime dividing $latex |G|$ then $latex Hunlhd G$. We have already seen a proof of this fact by considering the homomorphism $latex Gto S_p$ which is the induced map from $latex G$ acting on $latex G/H$ by left multiplication, and proving that $latex ker(Gto S_p)=H$. We now give an even shorter (and the just mentioned proof is already short) proof of this fact using double cosets.

$latex text{ }$

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