# Thread: two tricky group problems

1. ## two tricky group problems

(1)

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

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

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

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

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

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

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