# Thread: two tricky group problems

1. ## two tricky group problems

(1)

Prove that if $\displaystyle N$ is a normal subgroup of the finite group $\displaystyle G$ and $\displaystyle (|N|,|G:N|)=1$ then $\displaystyle N$ is the unique subgroup of $\displaystyle G$ of order $\displaystyle |N|$.
(2)

Let $\displaystyle K$ be a subgroup of some group (possibly infinite) $\displaystyle G$, and let $\displaystyle H$ be a subgroup of $\displaystyle K$. Prove that $\displaystyle |G:H|=|G:K|\cdot|K:H|$.
Regarding notation, $\displaystyle |G:H|$ means the number of left cosets of $\displaystyle H$ in $\displaystyle G$, that is, the number of elements in the set $\displaystyle \{gH:g\in G\}$. Also, $\displaystyle |N|$ is just the order of $\displaystyle N$, that is, the number of elements in $\displaystyle N$. Finally, $\displaystyle (a,b)$ denotes the greatest common divisor of the integers $\displaystyle a,b$.

I could probably get these with enough time, but I'm fresh out of it. Any hints (or heck, even just complete solutions) would be much appreciated!

2. For problem 1, since $\displaystyle G$ is finite, we have $\displaystyle |G:N|=|G|/|N|$, so the condition can be rephrased as $\displaystyle \gcd(|N|,|G|/|N|)=1$. I would try to prove by contradiction: suppose there is a another subgroup $\displaystyle N'$ such that $\displaystyle |N'|=|N|$. Then I would consider the quotient group $\displaystyle G/N$ and try mapping $\displaystyle N'$ into it, using the canonical homomorphism.

For problem 2, we can assume that $\displaystyle |G:K|$ and $\displaystyle |K:H|$ are finite. Let the collection of left cosets of $\displaystyle K$ in $\displaystyle G$ be $\displaystyle \{a_iK\ |\ i=1,2,\ldots r\}$ and the left cosets of $\displaystyle H$ in $\displaystyle K$ be $\displaystyle \{b_jH\ |\ j=1,2,\ldots s\}$. It would suffice to show that the set $\displaystyle \{a_ib_jH\ |\ i=1,2,\ldots r; j=1,2,\ldots s\}$ is the collection of left cosets of $\displaystyle H$ in $\displaystyle G$.

Can you give that a shot?

3. You may not have it in your group theory arsenal yet, but if you do,

Question 1. follows easily from Sylow's Theorem.

4. Thanks for the hints guys! I was able to get (1). However, problem (2) is still giving me trouble...

Originally Posted by roninpro
For problem 2, we can assume that $\displaystyle |G:K|$ and $\displaystyle |K:H|$ are finite. Let the collection of left cosets of $\displaystyle K$ in $\displaystyle G$ be $\displaystyle \{a_iK\ |\ i=1,2,\ldots r\}$ and the left cosets of $\displaystyle H$ in $\displaystyle K$ be $\displaystyle \{b_jH\ |\ j=1,2,\ldots s\}$. It would suffice to show that the set $\displaystyle \{a_ib_jH\ |\ i=1,2,\ldots r; j=1,2,\ldots s\}$ is the collection of left cosets of $\displaystyle H$ in $\displaystyle G$.
That's the approach I initially planned, but I just haven't had any luck getting the proof to fall together. Ordinarily I'd take a day or two away from it, and come back to it later with fresh eyes. Unfortunately, I'm nearly out of time, with only a couple hours to go.

5. The proof has two parts: you must show that (1) all of the cosets are distinct and (2) every left coset of $\displaystyle H$ in $\displaystyle G$ can be written in the form $\displaystyle a_ib_jH$.

For (1), what happens if $\displaystyle a_ib_jH=a_\alpha b_\beta H$?

For (2), what would it take for $\displaystyle gH=a_ib_iH$ (where $\displaystyle g\in G$)?