Homework Assignment 02

Math 210A, Fall 2026.

Problems to submit for grading

Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once you finish, 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!

Problem 1. 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 $p$-divisible1.

Hint: one possible approach is to follow the steps below:

  1. First, prove the result under the additional assumption that \(G\) is abelian. You want to show the map \(x\mapsto x^p\) is surjective, but since \(G\) is finite, this is the same as proving the map is injective, which should be easier to do. There are multiple ways to do this, but it might help to notice that the map is a homomorphism (since the group is abelian).
  2. To prove the result in general, take an arbitrary \(G\) and \(g\in G\), apply the first step to \(\langle g \rangle\), and tie things together.

Problem 2. Let \(G\) be a finite group. Prove that if \(G\) has a unique maximal2 subgroup, then $G$ is cyclic of prime-power order.

Hint: if \(M\) is the unique maximal subgroup and \(a\in G\setminus M\) what can you say about \(\langle a \rangle\)? You can use—without proof—the fact that every proper subgroup of a finite group is contained in some maximal subgroup.

Problem 3. Let \(n\) be a positive integer and \(Z_n\) be cyclic of order \(n\). For each \(k\in \mathbb{Z}\), define the function \(\sigma_k: Z_n \rightarrow Z_n\) by \(\sigma_k(x) = x^k\).

  1. Prove that \(\sigma_k\) is an automorphism of \(Z_n\) if and only if \(\gcd(k,n) = 1\).

    Hint: \(\sigma_k\) is quickly seen to be a homomorphism just using that \(Z_n\) is abelian. To study when it is surjective, recall that the image of a homomorphism (i.e. the range) is always a subgroup of the codomain, and in this case, the image of the homomorphism will be generated by the image of a generator for \(Z_n\). So you can investigate the order of the image of a generator to determine when the image of the homomorphism equals \(Z_n\). (Notice that whenever \(\sigma_k\) is surjective, injectivity follows immediately…see why?)

  2. Prove that if \(\varphi\) is any automorphism of \(Z_n\), then \(\varphi = \sigma_k\) for some integer \(k\).

    Hint: if \(\varphi\) is an automorphism and \(a\) is a generator of \(Z_n\), then what can you say about \(\varphi(a)\)?

Remark. What you’re proving in Problem 3 leads—with just a little more work—to the very important result that \(\operatorname{Aut}(Z_n) \cong (\mathbb{Z}/n\mathbb{Z})^\times\). See D&F Section 0.3 for the definition \((\mathbb{Z}/n\mathbb{Z})^\times\), which is also often referred to as \(U_n\).

Problems for extra practice

Please try to solve some of these problems, but you do not turn them in. I’m happy to talk about these problems (or any others) if you have questions.

Problem. Let \(G\) be a group. For elements \(x,y \in G\), define \([x,y] = x^{-1}y^{-1}xy\) (called the commutator of $x$ and $y$), and for subgroups \(H,K\le G\), define \([H,K] = \langle [h,k] \mid h\in H, k \in K \rangle\).

  1. Prove that \([H,K]=[K,H]\).
  2. Prove that \(H\le C_G(K)\) if and only if \([H,K]=1\).
  3. Prove that \(H\le N_G(K)\) if and only if \([H,K]\le K\).

Problem. Determine the maximal subgroups of \(D_8\).

Problems. work some problems from D&F Section 2.3. Here are some to consider: 8, 9, 12, 16, 18, 24.

Problems. work some problems from D&F Section 2.4. Here are some to consider: 11, 13, 14(c,d), 15, 16, 19

Footnotes

  1. Definition. Let \(n\) be a positive integer. A group $G$ is said to be $n$-divisible if for every \(g\in G\) there is some \(x\in G\) such that \(g=x^n\). Equivalently, $G$ is $n$-divisible if and only if the function \(G\rightarrow G:x\mapsto x^n\) is surjective.

    Remark. In other words, $G$ is $n$-divisible if every $g\in G$ has an $n^\text{th}$ root in $G$. And in additive notation, the condition $g=x^n$ becomes $g=nx$, justifying the name $n$-divisible. ↩

  2. Definition. A subgroup \(M<G\) is said to be a maximal subgroup of \(G\) if

    1. \(M\neq G\), and
    2. the only subgroups of \(G\) that contain \(M\) are \(M\) and \(G\) (i.e. \(M \le H \le G\) implies \(H=M\) or \(H=G\).)

    Remark. The maximal subgroups of a group are the largest of the proper subgroups. It is quite common for a group to have several maximal subgroups, and it is also possible for infinite groups to have none. ↩