Algebra

Tricky Algebra Problems:

If you're up for a bigger challenge or maybe you're just tired of doing a bunch of the standard exercises and reading your notes but still want to be productive, the following problems are for you! In no particular order, the following problems are a little more exotic and more than likely won't appear on your exams (although a few of them are from previous exams or directly out of Hungerford so no promises). Harder problems will be marked with a * symbol and even harder ones will be marked with a **.

1. * Let $$G$$ be a finite group with $$n^2 − n + 1$$ elements such that the map $$x \mapsto x^n$$ is an endomorphism of $$G$$. Prove that $$G$$ is abelian. (MathSE)

2. Prove that $$\text{Aut}(S_4) \cong S_4$$. Then prove that there is no group $$G$$ such that its commutator subgroup $$G'$$ is isomorphic to $$S_4$$.

3. * Prove that for any finite group $$G$$, there is some field $$K$$ and Galois extension $$F$$ over $$K$$ such that $$\text{Aut}_K(F) \cong G$$. This is similar to the rather famous open problem of, for any finite group $$G$$, finding a Galois extension $$F$$ of $$\mathbb{Q}$$ such that $$\text{Aut}_{\mathbb{Q}}(F) \cong G$$.

4. * Prove that the Galois group of $$K(x)$$ over $$K$$ is isomorphic to $$PGL_n(K)$$. (See Hungerford Chapter V, Section 3, Exercise 6)

5. For a prime integer $$p$$, let $$f$$ be an irreducible polynomial of degree $$p$$ over $$\mathbb{Q}$$ that has exactly two nonreal roots in $$\mathbb{C}$$. Then the Galois group of $$f$$ is the symmetric group $$S_p$$. (Hungerford Chapter V, Theorem 4.12)

6. * Prove that over any base field, the polynomial $$x^3 − 3x + 1$$ is either irreducible or splits completely. (MathSE)

7. ** Prove that every element of a finite field can be written as the sum of two squares. (MathSE)

8. Prove that if every proper ideal of a ring is prime, then that ring must be a field.

9. Prove that an infinite integral domain with finitely many units must have an infinite number of max ideals. In particular, the PID $$\mathbb{Z}$$ has infinitely many primes.

10. Recall that the Jacobson radical $$J$$ of a ring is the intersection of all max ideals in the ring. Prove that if $$x \in J$$, then $$x+1$$ is invertible. (MathSE)

11. Define the symmetric group $$S_n$$ as being the group of all automorphisms of the set $$\{1, \cdots, n\}$$. Prove that $$S_n$$ can be presented as being generated by $$\{s_1, \cdots, s_{n−1}\}$$ subject to the relations $s_i^2 = 1 \text{ for } i \in \{1, \cdots, n-1\}$ $(s_{i} s_{i+1})^3 = 1 \text{ for } i \in \{1, \cdots, n-2\}$ $s_i s_j = s_j s_i \text{ for } j \in \{1, \cdots, n-1\} \text{ and } i \neq j \pm 1.$

12. For a nontrivial free abelian group, find an subgroup of index $$n$$ for each $$n \in \mathbb{N}$$

13. * What is an example of a group that has a automorphism that isn’t an inner automorphism? (See this write up by Baez or this article by Vakil for a great example.)

14. ** Find an example of an abelian group $$A$$ such that $$A \cong A \oplus \mathbb{Z}^2$$ but $$A \not\cong A \oplus \mathbb{Z}$$. Use this example to construct a group $$G$$ such that $$G \cong G \oplus G \oplus G$$ but $$G \not\cong G \oplus G$$. (MathOverflow)

15. * Given an example of modules $$A$$, $$B$$, and $$C$$ such that $$B \cong A \oplus C$$ but the short exact sequence $0 \to A \to B \to C \to 0$ does not split. (MathSE)

16. * Recall that $$A \otimes_R B$$ is generated by decomposable (simple) tensors of the form $$a \otimes b$$ for $$a \in A$$ and $$b \in B$$. But in many basic examples, all the of the elements of $$A \otimes_R B$$ are decomposable tensors. What’s an example of a tensor product of modules that contains non-decomposable elements?(MathSE)

17. ** Recall that if $$F$$ is the splitting field of $$f \in K[x]$$ where $$\text{deg}(f) = n \geq 1$$, that $$[F:K] \leq n!$$. Can you come up with an example of a field $$K$$ and polynomial $$f$$ such that $$[F:K] = n!$$? (See Hungerford Chapter V, Section 4, Exercise 14)

18. Recall that if you consider the action of $$G$$ on itself by conjugation, this will partition $$G$$ into conjugacy classes, where $$x$$ and $$y$$ are in the same class when there is some $$g \in G$$ such that $$gxg^{−1} = y$$.
1. Classify all groups that have exactly two conjugacy classes. (MathSE)
2. * Classify all groups that have exactly three conjugacy classes. (MathSE)
3. ** Prove that for any positive integer $$n$$, there is an upper bound on $$|G|$$ where $$G$$ is a group with $$n$$ conjugacy classes. I.e. There are only finitely many groups with a given number of conjugacy classes.

19. * Let $$G$$ be an infinite group. Recall the construction of the group ring $$\mathbb{C}[G]$$. Prove that for $$a,b \in \mathbb{C}[G]$$, if $$ab = 1$$, then $$ba = 1$$. (MathSE)

20. Prove that $$R$$ is commutative given the following single condition
1. * $$x^3 = x$$ for all $$x \in R$$. (MathSE)
2. ** $$x^5 = x$$ for all $$x \in R$$. (MathSE)
3. ** For each $$x \in R$$ there is some $$n \in \mathbb{Z}_{\geq 2}$$ such that $$xn = x$$. (See the Jacobson density theorem)