Homework Assignment 06

Math 210A, Fall 2020.

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 3.3: 3, 8
Section 3.4: 4
Section 1.7: 2, 4, 8, 9, 11, 12

Additional Problem. Let $G$ be a group. Suppose that $H\le G$, $N\trianglelefteq G$, and $HN = G$. Prove that if $H$ and $N$ are $p$-groups, then $G$ is a $p$-group. (See the footnotes¹ for the definition of a $p$-group.)

Additional Problem. Find a composition series for each of $D_{10}$, $D_{12}$, $D_{16}$. Determine the composition factors in each case.

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!

Additional Problem. Use the First Isomorphism Theorem to prove each of the following group isomorphisms.
1. For $F$ a field, $\operatorname{GL}_n(F) / \operatorname{SL}_n(F) \cong F^\times$.
2. For $n=md$, $D_{2n}/\langle r^m \rangle \cong D_{2m}$.
Remember that $F^\times$ denotes $F -\{0\}$ as a group with respect to multiplication. To prove $G/N \cong H$, find a surjective homomorphism from $G$ to $H$ with kernel $N$.
Additional Problem. Let $G$ be a group. Suppose that $H\le G$, $N\trianglelefteq G$, and $HN = G$. Prove that if $H$ is abelian, then $G/N$ is also abelian. Conclude that if $H$ and $N$ are both abelian, then $G$ is solvable.
Additional Problem. Let $G$ be a group acting on a set $X$, and let $g\in G$. Prove that $C_G(g)$ acts on $\operatorname{Fix}(g)$ (via restriction of the original action to $C_G(g)$ and $\operatorname{Fix}(g)$).

The key point is to show that for all $c\in C_G(g)$ and all $x \in \operatorname{Fix}(g)$, $cx \in \operatorname{Fix}(g)$. And then the fact that $c\cdot (d\cdot x) = (cd)\cdot x$ for all $c,d\in C_G(g)$ and all $x\in \operatorname{Fix}(g)$ follows immediately from the fact that the equation actually holds for all $c,d\in G$ and all $x\in X$. (…do you see why?) See the footnotes² for the definition of $\operatorname{Fix}(g)$.

Footnotes

Definition. Let $p$ be a prime. A (possibly infinite) group $P$ is called a $p$-group if the order of every element of $P$ is a power of $p$; that is, $P$ is a $p$-group if for all $x\in P$, $|x| = p^k$ for some $k\in \mathbb{N}$. (This differs from the definition in Dummit and Foote, but we will later prove that the two definitions are equivalent for finite groups.) ↩
Definition. Let $G$ be a group acting on a set $X$. For $g\in G$, we define the fixed points of $g$ to be $\operatorname{Fix}(g) = \{ x \in X \mid g\cdot x = x\}$. ↩