Homework Assignment 03

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 2.3: 1, 3, 11, 12, 16, 26(a,d)
• Section 2.4: 7, 13, 14(c,d), 15

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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 Homework 03 to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

One of the problems below requires the following definition.

Definition. Let $n\in \mathbb{Z}$ be positive. A group $G$ is said to be $n$-divisible if for every $g\in G$ there is some $x\in G$ such that $g=x^n$. (In other words, $G$ is $n$-divisible if every $g\in G$ has an $n^\text{th}$ root in $G$.)

Remarks:

1. Notice how $G$ is $n$-divisible if and only if the function $G\rightarrow G:x\mapsto x^n$ is surjective.
2. In additive notation, the condition $g=x^n$ becomes $g=nx$, justifying the name $n$-divisible.

1. Section 2.3: 26(b,c)
   • You can use 26(a) when doing these parts, but you only turn in parts (b) and (c).
2. Additional Problem. Let $G$ be group. Suppose that $a,b\in G$ are commuting elements of relatively-prime finite orders. Prove that $|ab| = |a|\cdot |b|$ and that $\langle a,b\rangle = \langle ab\rangle$.
   • Don’t forget to make use of previous exercises…like #16 in $\S$ 2.3.
3. Additional Problem. Let $G$ be a finite group, and let $p$ be a prime. Prove that if $G$ has no elements of order $p$, then $G$ is $p$-divisible.

   Possible approach: (1) explain why it suffices to prove that every abelian subgroup of $G$ is $p$-divisible; (2) prove that if $H$ is an abelian subgroup of $G$, then $H$ is $p$-divisible (by studying the map $H\rightarrow H:x\mapsto x^n$).