querent

Let $𝐺\subset GL(𝑛,𝐾)$ be a finite $𝑝$-group, where $𝐾$ is a field of characteristic $𝑝$. ($𝐾$ is not assumed to be finite.) Show that $𝐺$ fixes a nonzero vector in its action on ${𝐾}^{𝑛}$. (Hint: consider first the action of the center of $𝐺$, and think about minimal polynomials of elements of $𝐺$.)

(i) Let $𝑅$ be a commutative Noetherian integral domain. Prove that any nonzero nonunit is a finite product of irreducible elements. (Here an irreducible is a nonzero nonunit $𝑝$ such that in any factorization $𝑝=𝑥𝑦$, $𝑥$ or $𝑦$ is a unit.)

(ii) Let $𝑅$ be a commutative Noetherian ring. Prove that every ideal $𝐼\ne 𝑅$ contains a finite product ${𝑃}_1\cdots {𝑃}_{𝑘}$ where ${𝑃}_{𝑖}$ are prime ideals satisfying $𝐼\subset {𝑃}_{𝑖}$.

Let $𝐴$ be a finitely generated integral domain over a field $𝑘$, and $𝐺$ a finite group acting on $𝐴$ as a $𝑘$-algebra.

(i) Prove that $𝐴$ is a finitely generated module over the $𝑘$-subalgebra ${𝐴}^{𝐺}$. (Hint: consider ${\prod }_{𝑔\in 𝐺}(𝑋-𝑔(𝑎))$ for $𝑎\in 𝐴$.)

(ii) Prove ${𝐴}^{𝐺}$ is a finitely generated $𝑘$-algebra.

Let $𝐺$ be a finite group and $𝐻$ a subgroup of index 2.

(i) Let $𝑉$ be an irreducible complex representation of $𝐺$. Show that the restriction of $𝑉$ to $𝐻$ either remains irreducible or is the direct sum of two non-isomorphic irreducible representations.

(ii) Assume that whenever two elements of $𝐻$ are conjugate in $𝐺$, they are conjugate in $𝐻$. Prove that the restriction of every irreducible complex representation of $𝐺$ to $𝐻$ remains irreducible.

Suppose the polynomial $𝑓(𝑋)={𝑋}^4+𝑎{𝑋}^2+𝑏\in \mathbb{Q}[𝑋]$ is irreducible over the rational numbers. Note the roots of $𝑓(𝑋)$ have the form $\pm 𝛼,\pm 𝛽$. Let $𝐸$ denote a splitting field of $𝑓(𝑋)$, and let $𝐺=Gal(𝐸/\mathbb{Q})$ be the Galois group of the polynomial $𝑓(𝑋)$.

(i) Show that $|𝐺|=4$ or 8 and if $|𝐺|=4$ then only the identity element of $𝐺$ fixes a root of $𝑓(𝑋)$.

(ii) Show that $𝐺$ is the direct product of two cyclic groups of order 2 if and only if $𝑏$ is a square in $\mathbb{Q}$.

(iii) Show that $𝐺$ is cyclic of order 4 if and only if $\frac{{𝑎}^2-4𝑏}{𝑏}$ is a square in $\mathbb{Q}$.

Let $𝑝$ be a prime.

(i) Construct a projective resolution of $\mathbb{Z}/𝑝\mathbb{Z}$ as a $\mathbb{Z}/{𝑝}^2\mathbb{Z}$-module, and use it to compute ${Ext}_{\mathbb{Z}/{𝑝}^2\mathbb{Z}}^{𝑖}(\mathbb{Z}/𝑝\mathbb{Z},\mathbb{Z}/𝑝\mathbb{Z})$ for $𝑖>0$.

(ii) Prove that $\mathbb{Z}/{𝑝}^2\mathbb{Z}$ is injective as a module over itself. Compute ${Ext}_{\mathbb{Z}/{𝑝}^2\mathbb{Z}}^{𝑖}(\mathbb{Z}/𝑝\mathbb{Z},\mathbb{Z}/𝑝\mathbb{Z})$ for $𝑖>0$ using a suitable injective resolution.

Let $𝐴$ be an $𝑛\times 𝑚$ matrix of integers, thus defining a linear map

Similarly, the transpose ${𝐴}^{𝑡}$ of $𝐴$ defines a linear map

Prove that the cokernel of ${𝑓}_{𝐴}$ and the cokernel of ${𝑓}_{{𝐴}^{𝑡}}$ have isomorphic torsion subgroups.

Let $𝐺={SL}_3({𝔽}_{𝑝})$, where $𝑝$ is an odd prime. Let $\ell$ be a prime divisor of ${𝑝}^2+𝑝+1$.

(i) Suppose $\ell >3$. Prove that the $\ell$-Sylow subgroups of $𝐺$ are cyclic.

(ii) Suppose that $\ell =3$. Prove that the $\ell$-Sylow subgroups of $𝐺$ are not cyclic.

Let $𝐺={SL}_{𝑛}({𝔽}_{𝑝})$ for a prime $𝑝$ and an integer $𝑛>1$.

(i) Find a Sylow $𝑝$-subgroup $𝑃$ of $𝐺$ and compute its order.

(ii) Give an explicit sequence of subgroups $1={𝑃}_0\subset {𝑃}_1\subset {𝑃}_2\subset \cdots \subset {𝑃}_{𝑚}=𝑃$ such that for all $0\le 𝑖<𝑚$, ${𝑃}_{𝑖}$ is normal in ${𝑃}_{𝑖+1}$ and the quotient ${𝑃}_{𝑖+1}/{𝑃}_{𝑖}$ is abelian.

Let $𝑅$ be a commutative ring, and $𝑀$ a free $𝑅$-module of finite rank $𝑛>0$.

(i) Let ${𝑀}^{\ast }$ be the dual module. For all $𝑖>0$ prove that ${Sym}^{𝑖}(𝑀)$ is free, and show that there is a unique bilinear pairing

such that

for all ${𝑚}_{𝑗}\in 𝑀$ and ${\ell }_{𝑗}\in {𝑀}^{\ast }$, where ${𝑆}_{𝑖}$ is the symmetric group on $\{1,\dots ,𝑖\}$.

(ii) Prove that the linear map ${Sym}^{𝑖}({𝑀}^{\ast })\to {Sym}^{𝑖}{(𝑀)}^{\ast }$ induced by the pairing in (i) is an isomorphism when $𝑖!\in {𝑅}^{\times }$ (hint: use a basis of $𝑀$ to make bases of these symmetric powers that are "dual" to each other up to ${𝑅}^{\times }$-multipliers), and for any prime $𝑝$ and $𝑀=𝑅={𝔽}_{𝑝}$ show that this map vanishes when $𝑖=𝑝$.

(iii) Let $0\to {𝑀}^{\prime }\to 𝑀\to {𝑀}^{″}\to 0$ be a short exact sequence of finitely generated free $𝑅$-modules, with $𝑀$ of rank $𝑛>0$. For any integer $0\le 𝑎\le 𝑛$, let ${𝑋}_{𝑎}\subset {\wedge }^{𝑛}(𝑀)$ be the $𝑅$-submodule generated by elementary wedge products involving at least $𝑎$ factors from ${𝑀}^{\prime }$ (and define ${𝑋}_{𝑛+1}=0$). Construct a natural isomorphism

for all $0\le 𝑎\le 𝑛$. State and prove a precise functorial property of this isomorphism that justifies calling it "natural".