Homework Assignment 07

Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 4.1: 1, 4, 10
• Section 4.2: 2, 9

Additional Problem. Let $G$ be an abelian group acting transitively on a set $X$. Prove that for all $g\in G$, if $gx = x$ for some $x\in X$, then $gx = x$ for all $x\in X$. Conclude that if the action is also faithful, then $|G| = |X|$.

Consider using the last additional problem from Homework 06.

Additional Problem. Let $H\le G$. Prove that if $N\trianglelefteq G$ and $N \le H$, then $N\le \operatorname{Core}_G(H)$. Conclude that $H\trianglelefteq G$ if and only if $H = \operatorname{Core}_G(H)$. [See the footnotes¹ for the definition of $\operatorname{Core}_G(H)$.]

Additional Problem. Let $G$ be an infinite group with a subgroup of finite index. Prove that $G$ has a normal subgroup of finite index, and conclude that $G$ is not simple.

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 4.1: 1

2. Additional Problem (The Core Lemma). Let $H\le G$ be a subgroup of finite index $n$. Prove that $|G: \operatorname{Core}_G(H)|$ is a divisor of $n!$ and a multiple of $n$.
   [See the footnotes¹ for the definition of $\operatorname{Core}_G(H)$.]

   To see that $|G: \operatorname{Core}_G(H)|$ is a divisor of $n!$, let $G$ act on $G/H$ by left multiplication. Remember from class that $\operatorname{Core}_G(H)$ is the kernel of the associated permutation representation. What can you say about the image of the associated permutation representation?

3. Additional Problem. Let $G$ be a group be a group of order $pq$ for primes $p< q$. Prove that $G$ has a normal subgroup of order $q$, and conclude that $G$ is not simple.
   You are encouraged to use Cauchy’s theorem², but please do NOT use Sylow’s theorems.

   Consider using the previous additional problem.

   Taking things a bit further: instead of just concluding that $G$ is not simple, prove also that $G$ is solvable.

   A generalization: once we prove Sylow’s theorems, we’ll be able to generalize this to show that groups of order $pq^k$ for primes $p< q$ always have a normal subgroup of order $q^k$.

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Footnotes