Algebra I - January 2025

January 2025 Algebra Qualifying Exam

Problem 1.

Let $G$ be a group of size $385=5\times7\times11$.

1. Show that $G$ contains normal subgroups of sizes 7 and 11, and the subgroup of size 7 is contained in the center $Z(G)$.

2. Show that there are only two possibilities for the group $G$ up to isomorphism.

Problem 2.

Let $R$ be a $\mathbb Q$-algebra that is finite-dimensional as a $\mathbb Q$-vector space.

1. Show that if $rs=1$, then $sr=1$ in $R$.

2. Show that if for any nonzero $x,y\in R$, we have $xy\ne0$, then $R$ is a division ring. (That is, every nonzero element has a multiplicative inverse.)

Problem 3.

Let $R$ be a commutative ring, and let $M$ be an $R$-module. Show that $M\simeq(0)$ if and only if $M_{\mathfrak m}=0$ for all maximal ideals $\mathfrak m$ in $R$. Recall that the notation $M_{\mathfrak m}$ means the localization of $M$ with respect to the multiplicative set $R\setminus\mathfrak m$.

Problem 4.

Let $G$ be a simple group, and let $H\subset G$ be a non-trivial proper subgroup.

1. Show that the subgroup $\widetilde H$ generated by all the conjugate subgroups $gHg^{-1}$, for $g\in G$, is equal to $G$ itself.

2. Show that the union of all the conjugate subgroups $gHg^{-1}$, for $g\in G$, is not a subgroup, assuming that $G$ is finite.

3. Find a counterexample to part (b) assuming that $G$ is infinite. Hint: Matrix groups. No justification is required for the counterexample.

Problem 5.

Let $R$ be a commutative ring, and let $x\in R$ be an idempotent element ($x^2=x$) with $x\ne0,1$. Show that $R\simeq R_1\times R_2$ for some nonzero rings $R_1$ and $R_2$.

Problem 6.

Let $R$ be a commutative ring, and let $M$ be an $R$-module. For a given submodule $N\subset M$, let

$$\operatorname{Ann}(N)=\{r\in R:r\cdot n=0\text{ for all }n\in N\}.$$

1. Show that $\operatorname{Ann}(N)\subset R$ is an ideal.

2. Let $V$ be a finite-dimensional $F$-vector space, and let $T:V\to V$ be a $F$-linear map. Viewing $V$ as an $F[t]$-module in the usual way, show that for any polynomial $q(t)\in F[t]$ dividing the minimal polynomial of $T$, there exists an $F[t]$-submodule $W\subset V$ such that $\operatorname{Ann}(W)=(q(t))$.