A Clever Proof of a Common Fact
Point of Post: In this post we give a new proof that if is a finite group is a subgroup whose index is the smallest prime dividing then that subgroup is normal.
It is a commonly used theorem in finite group theory that if is a finite group and such that is the smallest prime dividing then . We have already seen a proof of this fact by considering the homomorphism which is the induced map from acting on by left multiplication, and proving that . We now give an even shorter (and the just mentioned proof is already short) proof of this fact using double cosets.
We recall that a double coset for subgroups is an orbit of the action of on given by . We also recall that . With this we can easily prove the theorem
Theorem: Let be a finite group and the smallest prime dividing . Then, if is such that then .
Proof: We know that decomposes as
where . We see then that that
Now, we know then that and since it’s a divisor of we evidently mus have that and so . Since we can evidently choose arbritrarily in (since it is a representative for one of the double cosets) the conclusion follows.
1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2009
No comments yet.