- #1

Whovian

- 652

- 3

## Homework Statement

While reading through my textbook on abstract algebra while studying for a test, I ran across the following statement:

There are two isomorphism classes of groups of order 21: the class of ##C_{21}##, and the class of a group ##G## generated by two elements ##x## and ##y## that satisfy the relations ##x^7=1##, ##y^3=1##, ##yx=x^2y##

along with a proof of this statement. The proof looks fairly reasonable, only there's a bit I don't quite understand:

"The Third Sylow Theorem shows us that the Sylow 7-subgroup ##K## [existence and uniqueness guaranteed by the Third Sylow Theorem] must be normal ..."

Specifically, why must ##K\trianglelefteq G##?

## Homework Equations

Third Sylow Theorem: If ##G## is a group of order ##n<\aleph_0##, where ##n=p^e\cdot m## for some ##e>0## and ##\lnot\left(p|m\right)##, for some prime ##p##, and ##s## is the number of Sylow ##p##-subgroups of ##G##, then ##s|m## and ##s\equiv1\pmod p##.

## The Attempt at a Solution

Fairly nonexistent; I can't think of any reason this subgroup must be normal.

(If this post would be more appropriate in the linear & abstract algebra section, feel free to move it there.)