Exercises

Group Theory

Suppose that $G$ has a subgroup $H$ of finite index. Prove that the number of left cosets of $H$ in $G$ is the same as the number of right cosets.

Consider a finite group $G$ and $N \triangleleft G$.
1. For $g∈G$, prove that the order of $gN$ in $G/N$ divides the order of $g$ in $G$.
2. For a subgroup $H$ of $G$, prove that if $|H|$ and $[G:N]$ are relatively prime, then $H$ is a subgroup of $N$.
3. Prove that if $|N|$ and $[G:N]$ are relatively prime, then $N$ is the only subgroup of $G$ with order $|N|$.

Prove that a subgroup of index two must be normal.

Can you find examples of a groups $K$, $H$, and $G$ such that $H \triangleleft G$ and $K \triangleleft H$, but $K$ is not normal in $G$?

What's a finite group G with normal subgroups A and B such that
1. $A \cong B$ but $G/A \ncong G/B$?
2. $A \ncong B$ but $G/A \cong G/B$?

Prove that every (nontrivial) subgroup of $\mathbb{Z}$ is cyclic.

Let $G$ be an abelian group, and let $H$ be a subgroup of $G$. Prove that if there is a homomorphism $ \varphi : G \to H$ such that $\varphi$ restricted to $H$ is the identity, then $G \cong H \times \ker\varphi$.

Suppose that $\sigma \in S_n$ is given in cyclic notation as $(i_1 i_2 \cdots i_m)$. For $\tau \in S_n$ prove that $\tau\sigma\tau^{-1} = (\tau(i_1)\tau(i_2) \cdots \tau(i_m))$.

An element of $S_n$ may be written as a product of transpositions. Among all such ways of writing an element as a product of transpositions, there is a minimal number of transpositions necessary to write that element. For any $n$, what is the maximum number over all elements of $S_n$ of this minimal number of transpositions that you need to write that element?

For a finitely generated abelian group $G$, recall the definition of the invariant factors of $G$ and of the elementary divisors of $G$. For an abelian group of the following orders, write down every possibility for its list of invariant factors and elementary divisors.\[165 \quad 180 \quad 128\]

Prove that if $G$ is a finite noncyclic abelian group, then $Aut(G)$ is not abelian.

For a group $G$ acting on a set $X$, for an element $x \in X$ recall the definition of the orbit of $x$, denoted $G.x$, and of the stabilizer of $x$, denoted $G_x$. Prove that $|G.x| = [G:G_x]$.

What is the definition of an inner automorphism of a group? Prove that the group of inner automorphisms of a group $G$ form a normal subgroup of $Aut(G)$. Furthermore prove that the group of inner automorphisms is isomorphic to $G/Z(G)$, where $Z(G)$ denotes the center of $G$.

Prove that if a group contains an element of order greater than two, then it must have a nontrivial automorphism.

Prove that $G/Z(G)$ is cyclic if and only if $G$ is abelian.

Related to the previous question, prove that $Aut(G)$ being cyclic means $G$ is abelian. What's an example of an abelian group with non-cyclic automorphism group? (MathSE)

For a group $G$ and a subgroup $H$ of $G$ of finite index, prove that there must exist a normal subgroup $N$ of $G$ contained in $H$ that also has finite index. (MathSE)

A variation on the previous exercise: If $G$ is a finite simple group with a subgroup $H$ of index $n$, show that $G$ is isomorphic to a subgroup of $S_n$.

For a finite group $G$ with subgroup $H$ of index $p$, if $p$ is the smallest prime divisor of $|G|$, then $H$ must be normal in $G$. (MathSE)

Prove that a finite $p$-group has nontrivial center.

Prove that if $|G|=p^n$ for some prime integer $p$, then $Z(G)$ is nontrivial.

Prove that for a normal Sylow $p$-subgroup $P$ of a finite group $G$, and an endomorphism $\phi$ of $G$, that $\phi(P)$ is a subgroup of $P$. Is this true if $G$ is infinite?

Prove that if $|G| = p^2$ for some prime integer $p$, then $G$ is abelian.

Suppose that $p$ and $q$ are prime integers such that $p > q$. Prove that if $|G|=p^n q$, then $G$ cannot be simple.

Show that a group of any of the following orders cannot be simple. (MathSE)\[105 \quad 120 \quad 200 \quad 250\]

For a group $G$, what is the definition of its commutator subgroup? Denote the commutator subgroup as $G'$. Prove that $G'$ is normal in $G$, and show that for any abelian group $A$, a homomorphism $G \to A$ must factor through the quotient $G/G'$.

Recall what it means for a group to be nilpotent and what it means for a group to be solvable. Prove that a nilpotent group is solvable. (See Hungerford Chapter II, Section 7, Exercises 3 and 4 for an different characterizations of solvability and nilpotency that make this proof easier.)

Prove that every subgroup and every homomorphic image of a solvable group is solvable.

If $N$ is normal in $G$ and both $N$ and $G/N$ are solvable, prove that $G$ is solvable too.

Ring Theory

Prove that a finite integral domain is in fact a field.

Recall what it means for an element of a ring to be nilpotent. For a commutative unital ring $R$, prove that the set of nilpotent elements forms an ideal.

Prove that if $R$ is commutative and both $a$ and $b$ in $R$ are nilpotent, the their sum $a + b$ is nilpotent. Why do we need $R$ to be commutative?

Let $R$ be a field of characteristic $p \neq 0$. Show that the Frobenius map ($r \mapsto r^p$) is an isomorphism

What is an example of an integral domain $R$ and ideals $I$ and $J$ such that $IJ \neq I \cap J$?

In a commutative unital ring, prove that maximal ideas are prime. Prove that the converse is true if your ring is a PID.

In the category of commutative unital rings, give an example of a
1. Ring that is not an Integral Domain.
2. Integral Domain that is not a GCD Domain.
3. Integral Domain that is not a UFD. (Bonus points if your example is a GCD Domain.)
4. UFD that is not a PID.
5. PID that is not a Euclidian Domain.
6. Euclidean Domain that is not a Field.

For a commutative unital ring $R$, an ideal $M$ is maximal if and only if for each $r \in R\setminus M$ there is some $s \in R$ such that $1 − rs \in M$.

Recall the definition of an idempotent element of a ring and of a central element of a ring. Two elements $a$ and $b$ of a ring are orthogonal if $ab = 0$. If $R$ is a unital ring with idempotent element $e$,
1. then the element $1-e$ is also idempotent,
2. and if $e$ is a central element of $R$, then $eR$ and $(1−e)R$ are ideals such that $R = eR \times (1-e)R$.
3. More generally, there are ideals $\{J_i\}_{i \in 1, \cdots, n}$ of $R$ such that $R$ can be written as an internal direct sum of the $J_i$, i.e. $R = J_1 \oplus \cdots \oplus J_n$, if and only if $R$ contains orthogonal central idempotents $\{e_i\}_{i \in 1, \cdots, n}$ such that $e_1 + \cdots + e_n = 1$ and $J_i = e_i R$ for $i \in \{1,…,n\}$. This is called the Peirce decomposition of a ring.

Recall the definition of a local ring. Prove that a commutative unital ring $R$ is local if and only if for all $a,b \in R$ we have that $a + b = 1$ implies that either $a$ or $b$ is a unit.

Prove that $R$ is local if every non-unit of $R$ is nilpotent.

For a unital ring $R$ of characteristic $p$, let $a$ be a nilpotent element of $R$. Prove that $a + 1$ is unipotent (that some power of $a + 1$ equals $1$).

What’s an example of an integral domain $R$ with non-maximal ideal $I$ such that $char R = 0$ but $char R/I \neq 0$?

For a commutative unital ring $R$, suppose that $f = a_nx^n + a_{n−1}x^{n−1} + \cdots + a_0$ is a zero divisor in $R[x]$. Prove that there exists some $b \in R$ such that $b a_n = b a_{n-1} = \cdots = b a_0 = 0$.

For a commutative unital ring $R$ and polynomial $f = a_nx^n + a_{n−1}x^{n−1} + \cdots + a_0 \in R[x]$, $f$ is a unit in $R[x]$ if and only if $a_0$ is a unit in $R$ and $a_1, \cdots, a_n$ are nilpotent.

Suppose $R$ is a ring such that $r^2 = r$ for all $r \in R$. Prove
1. every element is its own additive inverse.
2. $R$ is commutative.
3. every finitely generated ideal is principal. (MathSE)
4. every prime ideal is maximal.

For indeterminates $x$ and $y$ and a field $k$, prove that $(x,y)$ is not a principal ideal of $k[x,y]$.

Commutative Algebra

Prove that these three characterizations of $\text{Rad}(I)$, the radical of an ideal $I$ of a commutative unital ring $R$, are equivalent. The first one is the usual definition.
1. $\text{Rad}(I) = \{r \in R : r^n \in I, n \in \mathbb{N}\}$
2. $\text{Rad}(I)$ is the intersection of all prime ideals of $R$ that contain $I$.
3. $\text{Rad}(I)$ is the pre-image of the ideal of nilpotent elements in $R/I$.
It would be a good idea to prove that $\text{Rad}(I)$ is an honest ideal of $R$ directly from the first of these characterizations.

For a multiplicative subset $S$ of a commutative unital ring $R$, and an ideal $I$ of $R$, prove that $S^{-1} \text{Rad}(I) = \text{Rad}(S^{-1}I)$.

What’s an example of a Noetherian integral domain that is not a PID?

For a commutative unital ring $R$, let $I$ be a primary ideal of $R$, which means that for $a,b \in R$ such that $ab \in I$, either $a \in I$ or $b^n \in I$ for some $n \in N$. Let $S$ be a multiplicative subset of $R$ such that $S \cap I = \emptyset$. Prove that $S^{-1}I$ is a primary ideal of $S^{−1}R$.

For a commutative unital ring $R$ and proper ideal $I$ of $R$, prove that $I$ is a primary ideal if and only if the zero divisors in $R/I$ are all nilpotent.

For a commutative unital ring $R$, let $S$ be a saturated multiplicative subset $R$, so for $x,y \in R$ we have that if $xy \in S$ then $x,y \in S$. Prove that $R \setminus S$ is a union of prime ideals of $R$

For a commutative unital ring $R$, prove that the set of zero divisors of $R$ is a union of prime ideals.

Modules

Recall what it means for arbitrary extension of rings, $A \subset B$, to be finite and free.
1. Let $\{b_1, \cdots, b_n\}$ be a free basis for $A \subset B$. If $b = a_1b_1 + \cdots + a_nb_n$ for each $a_i \in A$ and some $a_s$ is a unit, show one may replace $b_s$ with $b$ to obtain another free basis.
2. Show that the natural inclusion $\mathbb{Q}[x] \hookrightarrow \mathbb{Q}[x,y]/\langle y^3 - x^2 y^2 + 7xy - 11x^{100} \rangle$ is finite and free.
3. Show that $\mathbb{F}_p[x^p] \subset \mathbb{F}_p[x]$ is finite and free. What's the canonical free basis for this extension?
4. Let $R = \mathbb{F}_p[x_1, \cdots, x_n]$. Then each extension in $\cdots \subset R^{p^k} \subset \cdots \subset R^{p^2} \subset R^p \subset R \subset R^{1/p} \subset \cdots$ is finite and free.

For unital ring $R$, recall what it means for a unitary $R$-module to be simple. Prove that a simple $R$-module $M$ must be cyclic, and that the ring $\text{End}_R(M)$ is a division ring. What about the converse? Is it true that if $\text{End}_R(M)$ is a division ring then $M$ must be simple?

For a commutative unital ring $R$ and left $R$-modules $M$ and $N$, does $\text{Hom}_R(M,N)$ have any sort of $R$-module structure? Is it necessary to assume that $R$ is commutative? What if $M$ is a right $R$-module instead? (MathSE)

For a ring $R$, consider the commutative diagram $$\require{AMScd} \begin{CD} 0 @>>> A @>{f_1}>> B @>{f_2}>> C @>>> 0\\ @. @. @V{\phi_2}VV @. @.\\ 0 @>>> X @>{g_1}>> Y @>{g_2}>> Z @>>> 0\\ \end{CD}$$ in the category of $R$-modules such that the top and bottom rows are exact. Suppose that there is a some map $\phi_1 \in \text{Hom}_R(A,X)$ such that $\phi_2 \circ f_1 = g_1 \circ \phi_1$. Prove that there exists some map $\phi_3 \in \text{Hom}_R(C,Z)$ such that $\phi_3 \circ f_2 = g_2 \circ \phi_2$.

Suppose that $P$ is a projective $R$-module, and is the homomorphic image of some $R$-module $M$. Prove that $P$ is isomorphic to a direct summand of $M$. What is the analogous fact to this one concerning injective $R$-modules?

For a unital ring $R$, in the category $R-\text{Mod}$, a free module is projective.

More generally than the previous problem, consider the three following adjectives that could describe an $R$-module: $\text{free} \quad \text{projective} \quad \text{torsion-free}$
Which of these properties of an $R$-module imply another, and which don't? Provide proofs and counterexamples.

Prove that a direct sum of $R$-modules $\oplus_{i \in I} P_i$ is projective if and only if each $P_i$ is projective.

Prove that $\mathbb{Q}$ is not a projective $\mathbb{Z}$-module. What is an example of a projective $\mathbb{Z}$-module?

Recall the definition of a $\mathbb{Z}$-module (abelian group) being divisible. Prove that a unitary $\mathbb{Z}$-module is injective if an only if it is divisible.

Suppose that in the category $R$-mod, for any object $D$ the functor $\text{Hom}_R(D,−)$ preserves the exactness of the sequence \[0 \to A \to B \to C \to 0.\] Prove that this sequence must split. Prove the converse of this statement too.

For a unital ring $R$ and a unitary left $R$-module $M$, write out the details of the left $R$-module isomorphism $M \cong \text{Hom}_R(R,M)$.

For a left $R$-module $M$, write down the details of the natural homomorphism of $R$-modules $\theta_M : A \to A^{**}$. Prove that $\theta_M$ is an isomorphism if $R$ is unital and $M$ is free with finite basis over $R$.

For a homomorphism of left $R$-modules $f: M \to N$, write down the details of the natural map $f^∗: M^{∗∗} \to N^{∗∗}$ such that the following diagram commutes: $$ \require{AMScd} \begin{CD} M @>{\theta_M}>> M^{\ast\ast}\\ @V{f}VV @VV{f^\ast}V\\ N @>{\theta_N}>> N^{\ast\ast}\\ \end{CD} $$

For a unital ring $R$ and a unitary left $R$-module $M$, write out the details of the left $R$-module isomorphism $R \otimes_R M \cong M$.

Let $R$ be a PID and suppose $V$ and $W$ are (not fully torsion) finite dimensional modules over $R$. Show the following:
1. $\text{dim}(V \oplus W) = \text{dim}(V) + \text{dim}(W)$
2. $\text{dim}(V \otimes W) = \text{dim}(V) \text{dim}(W)$

For integers $m$ and $n$, write out the details of the $\mathbb{Z}$-bimodule isomorphism $\mathbb{Z}/(m) \otimes_\mathbb{Z} \mathbb{Z}/(n) \cong \mathbb{Z}/(m,n)$. (MathSE)

Let $S$ be a two-sided ideal of a ring $R$ and let $SM$ denote the abelian subgroup of an $R$-module $M$ generated by elements of the form $sm$ for $s \in S$ and $m \in M$. Show that $SM$ is an honest submodule of $M$, describe the natural left $R$-module structure on $(R/S) \otimes_R M$, and show that $(R/S)\otimes_R M \cong M/SM$ as left $R$-modules.

Suppose that $A$ and $A'$ are left $R$-modules and $B$ and $B'$ are right $R$-modules. Take $f \in Hom(A,A')$ and $g \in \text{Hom}(B,B')$. Is it necessarily true that \[\ker(f \otimes g) \cong (\ker f \otimes B) + (A \otimes \ker g)?\]

Give examples of a commutative ring $R$, of $R$-modules $M$, $M'$, and $N$, and of a map $f \in \text{Hom}(M,M')$ such that
1. $f$ is injective, but $1 \otimes f : N \otimes M \to N \otimes M'$ is not injective.
2. $f$ is surjective, but $f^∗ :\text{Hom}(N,M) \to \text{Hom}(N,M')$, where $f^∗(h) = f \circ h$, is not surjective.

For a ring $R$ and left $R$-modules $M$ and $N$, write down the details of the homomorphism of abelian groups $M^∗ \otimes_R N \to \text{Hom}_R(M,N)$. Prove that this homomorphism is an isomorphism if $R$ is a field and $M$ and $N$ are finite-dimensional vector spaces over $R$.

Let $R$ be an integral domain. For an $R$-module $M$, define $\tau(M) = \{m \in M : \mathcal{O}_m \neq \emptyset\}$, where $\mathcal{O}_m$ is the annihilator of $m$ in $R$. Prove that $\tau$ induces a left-exact functor from $R$-mod to the category of torsion $R$-modules, where $M \mapsto \tau(M)$ and $f \mapsto f\mid_{\tau(M)}$. Why do we need the assumption that $R$ is an integral domain?

Let $R$ be a PID, and let $M$ be a unitary left $R$-module. For $s \in R$ recall the definition of a couple of our favorite submodules of $M$: $sM = \{sm : m \in M\}$ and $M[s] = \{ m \in M : sm = 0 \}$. Let $p$ be a prime element of $R$. Additionally, recall the definition of a cyclic $R$-module, and let $N$ be a cyclic $R$-module of order $r \in R$.
1. What is the natural way to define $M/pM$ as a vector space over $R/(p)$?
2. What is the natural way to define $M[p]$ as a vector space over $R/(p)$?
3. Supposing $s$ is relatively prime to $r$, prove that $sN = N$ and $N[s] = 0$.
4. Suppose $s$ divides $r$, so there is some $k$ such that $sk = r$. Prove that $sN \cong R/(k)$ and $N[s] \cong R/(s)$.

Linear Algebra

For a division ring $D$, let $V_i$ be a finite dimensional vector space over $D$ for $i \in \{ 1, \cdots, k\}$. Suppose the sequence \[0 \to V_1 \to V_2 \to \cdots \to V_k \to 0\] is exact. Prove that $\sum_{i=1}^{k} (-1)^{-i}\text{dim}_D (V_i) = 0$

Prove that if $A$ and $B$ are invertible matrices over a field $k$, then $A + \lambda B$ is invertible for all but finitely many $\lambda \in k$.

For the ring of $n \times n$ matrices over a commutative unital ring $R$, which we'll denote $\text{Mat}_n(R)$, recall the definition of the determinant map $\text{det}: \text{Mat}_n(R) \to R$. For $A \in \text{Mat}_n(R)$ also recall the definition of the classical adjoint $A^\alpha$ of $A$
1. $\text{det}(A^\alpha) = \text{det}(A)^{n-1}$
2. $(A^\alpha)^\alpha = \text{det}(A)^{n-2}A$

If $R$ is an integral domain and $A$ is an $n \times n$ matrix over $R$, prove that if a system of linear equations $Ax = 0$ has a nonzero solution then $\text{det}A = 0$. Is the converse true? What if we drop the assumption that $R$ is an integral domain?

What is the companion matrix $M$ of the polynomial $f = x^2 - x + 2$ over $\mathbb{C}$? Prove that $f$ is the minimal polynomial of $M$.

Suppose that $\phi$ and $\psi$ are commuting endomorphisms of a finite dimensional vector space $E$ over a field $k$, so $\phi\psi = \psi\phi$.
1. Prove that if $k$ is algebraically closed, then $\phi$ and $\psi$ have a common eigenvector.
2. Prove that if $E$ has a basis consisting of eigenvectors of $\phi$ and $E$ has a basis consisting of eigenvectors of $\psi$, then $E$ has a basis consisting of vectors that are eigenvectors for both $\phi$ and $\psi$ simultaneously.

Galois Theory

Suppose that for an extension field $F$ over $K$ and for $a \in F$, we have that $b \in F$ is algebraic over $K(a)$ but transcendental over $K$. Prove that $a$ is algebraic over $K(b)$.

Suppose that for a field $F/K$ that $a \in F$ is algebraic and has odd degree over $K$. Prove that $a^2$ is also algebraic and has odd degree over $K$, and furthermore that $K(a) = K(a^2)$.

For a polynomial $f \in K[x]$, prove that if $r \in F$ is a root of $f$ then for any $\sigma \in \text{Aut}_K(F)$, $\sigma(r)$ is also a root of $f$.

Prove that as extensions of $\mathbb{Q}$, $\mathbb{Q}(x)$ is Galois over $\mathbb{Q}(x^2)$ but not over $\mathbb{Q}(x^3)$.

If $F$ is _______ over $E$, and $E$ is _______ over $K$, is $F$ necessarily _______ over $K$? Answer this question for each of the words "algebraic," "normal," and "separable" in the blanks.

If $F$ is _______ over $K$, and $E$ is an intermediate extension of $F$ over $K$, is $F$ necessarily _______ over $E$? Answer this question for each of the words "algebraic", "normal", and "separable" in the blanks.

If $F$ is some (not necessarily Galois) field extension over $K$ such that $[F:K] = 6$ and $\text{Aut}_K(F) \cong S_3$, then $F$ is the splitting field of an irreducible cubic over $K[x]$.

Recall the definition of the join of two subgroups $H \vee G$ (or $H + G$). For $F$ a finite dimensional Galois extension over $K$ and let $A$ and $B$ be intermediate extensions. Prove that
1. $\text{Aut}_{AB}(F) = \text{Aut}_A(F) \cap \text{Aut}_B(F)$
2. $\text{Aut}_{A \cap B}(F) = \text{Aut}_A(F) \vee \text{Aut}_B(F)$

For a field $K$ take $f \in K[x]$ and let $n = \text{deg}(f)$. Prove that for a splitting field $F$ of $f$ over $K$ that $[F:K] \leq n!$. Furthermore prove that $[F:K]$ divides $n!$.

Let $F$ be the splitting field of $f \in K[x]$ over $K$. Prove that if $g \in K[x]$ is irreducible and has a root in $F$, then $g$ splits into linear factors over $F$.

Prove that a finite field cannot be algebraically closed.

For $u = \sqrt{2 + \sqrt{2}}$, what is the Galois group of $\mathbb{Q}(u)$ over $\mathbb{Q}$? What are the intermediate fields of the extension $\mathbb{Q}(u)$ over $\mathbb{Q}$?

Characterize the splitting field and all intermediate fields of the polynomial $(x^2 - 2)(x^2 - 3)(x^2 - 5)$ over $\mathbb{Q}$. Using this characterization, find a primitive element of the splitting field.

Characterize the splitting field and all intermediate fields of the polynomial $x^4 - 3$ over $\mathbb{Q}$

Consider the polynomial $f = x^3 − x + 1$ in $\mathbb{F}_3[x]$. Prove that $f$ is irreducible. Calculate the degree of the splitting field of $f$ over $\mathbb{F}_3$ and the cardinality of the splitting field of $f$.

Given an example of a finite extension of fields that has infinitely many intermediate fields.

Let $u = \sqrt{3 + \sqrt{2}}$. Is $\mathbb{Q}(u)$ a splitting field of $u$ over $\mathbb{Q}$? (MathSE)

Prove that the multiplicative group of units of a finite field must be cyclic, and so is generated by a single element.

Prove that $\mathbb{F}_{p^n}$ is the splitting field of $x^{p^n} - x$ over $\mathbb{F}_p$.

Prove that for any positive integer $n$ there is an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

Recall the definition of a perfect field. Give an example of an imperfect field, and the prove that every finite field is perfect.

For $n > 2$ let $\zeta_n$ denote a primitive $n$th root of unity over $\mathbb{Q}$. Prove that \[[\mathbb{Q}(\zeta_n + \zeta_n^{-1} : \mathbb{Q})] = \frac{1}{2}\varphi(n)\] where $\varphi$ is Euler's totient function.

Suppose that a field $K$ with characteristic not equal to $2$ contains an primitive $n$th root of unity for some odd integer $n$. Prove that $K$ must also contain a primitive $2n$th root of unity.

Prove that the Galois group of the polynomial $x^n - 1$ over $\mathbb{Q}$ is abelian. (MathSE)