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Let $𝐺$ is a group of order 16 generated by two elements $𝑥$ and $𝑦$ which satisfy ${𝑥}^8=1$ and ${𝑦}^2=1$ and $𝑦𝑥𝑦={𝑥}^3$.

(a) Prove that the derived group ${𝐺}^{\prime }$ generated by commutators is the cyclic group $⟨{𝑥}^2⟩$ of order 4.

(b) How many irreducible representations does $𝐺$ have and what are their degrees?

(c) For each irreducible character $𝜃$ of $𝐺$ of degree greater than 1 compute $𝜃(𝑥)$ and $𝜃(𝑦)$.

(a) Show that ${𝐴}_6$ has no subgroup of index 3. (Hint: you could for instance think about elements of order 5.)

(b) Show that there is no simple group of order 120.

Let $(𝑅,𝔪)$ be a noetherian local ring with residue field $𝑘=𝑅/𝔪$, and $𝑀$ a finitely generated $𝑅$-module.

(a) Let $𝑓:{𝑅}^{𝑛}\to 𝑀$ be the linear map defined by ${𝑚}_1,\dots ,{𝑚}_{𝑛}\in 𝑀$ lifting a $𝑘$-basis of ${𝑀}_0=𝑀/𝔪𝑀$. Prove $𝑓$ is surjective and $𝐾=ker(𝑓)$ is finitely generated.

(b) Prove that if $𝑀$ is $𝑅$-flat then $𝑓$ is an isomorphism (so $𝑀$ is free).

Let $𝐺$ be a finite group, and let $𝜃:𝐺\to 𝐺$ be an automorphism of order 2. Also let $𝑉$ and $𝑊$ be irreducible $ℂ[𝐺]$-modules that are finite dimensional over $ℂ$.

(a) Prove that the space of bilinear maps $𝛽:𝑉\times 𝑊\to ℂ$ such that $𝛽(𝑔𝑣,𝜃(𝑔)𝑤)=𝛽(𝑣,𝑤)$ for all $𝑔\in 𝐺$, $𝑣\in 𝑉$, $𝑤\in 𝑊$ is at most one-dimensional.

(b) If $𝛽:𝑉\times 𝑉\to ℂ$ is a non-zero bilinear form such that $𝛽(𝑔𝑣,𝜃(𝑔)𝑤)=𝛽(𝑣,𝑤)$, show that $𝛽$ is non-degenerate.

(c) If $𝛽:𝑉\times 𝑉\to ℂ$ is a non-zero bilinear form such that $𝛽(𝑔𝑣,𝜃(𝑔)𝑤)=𝛽(𝑣,𝑤)$ show that either $𝛽(𝑣,𝑤)=𝛽(𝑤,𝑣)$ for all $𝑣,𝑤\in 𝑉$ or $𝛽(𝑣,𝑤)=-𝛽(𝑤,𝑣)$ for all $𝑣,𝑤\in 𝑉$.

Let $𝑘$ be an algebraically closed field and let $𝑛\in {\mathbb{Z}}_{\ge 2}$. Let $𝐼\subset 𝑘[{𝑥}_1,\dots ,{𝑥}_{𝑛},{𝑦}_1,\dots ,{𝑦}_{𝑛}]$ be the ideal generated by $({𝑥}_{𝑖}{𝑦}_{𝑗}-{𝑥}_{𝑗}{𝑦}_{𝑖})$ $(1\le 𝑖<𝑗\le 𝑛)$. Informally, it corresponds to the matrix

having rank at most 1. Let $𝑉(𝐼)\subset {𝑘}^{2𝑛}$ be the vanishing locus of $𝐼$. Show that $𝑉(𝐼)$ is irreducible in the Zariski topology and find its dimension in terms of $𝑛$. (Hint: one possible approach would be to look at the loci where ${𝑥}_{𝑖}\ne 0$ for some $𝑖$ or ${𝑦}_{𝑗}\ne 0$ for some $𝑗$.)

Suppose that $𝐴$ is a noetherian integral domain with field of fractions $𝐾$. We call $𝑎\in 𝐾$ almost integral over $𝐴$ if there exists $𝑏\in 𝐴-\{0\}$ with $𝑏{𝑎}^{𝑖}\in 𝐴$ for all $𝑖\in {\mathbb{Z}}_{>0}$.

(a) Show that $𝑎\in 𝐾$ is integral over $𝐴$ if and only if it is almost integral.

(b) If $𝑓\in 𝐾[𝑋]$ is integral over $𝐴[𝑋]$, show that the coefficients of $𝑓$ are integral over $𝐴$. (Hint: you might like to use induction on the degree of $𝑓$ and part (a).)

(c) Show that if $𝐴$ is integrally closed, then $𝐴[𝑋]$ is also integrally closed. (Hint: first treat the case that $𝐴$ is a field.)

(d) Suppose that the integral closure of $𝐴$ in its field of fractions is a finitely generated $𝐴$-module. Prove that the integral closure of $𝐴[𝑋]$ in its field of fractions is a finitely generated $𝐴[𝑋]$-module.

State a theorem which classifies (i.e. lists) all finite abelian groups up to isomorphism. This means that each finite abelian group should be isomorphic to exactly one group of your list. Use your classification to list abelian groups of order $24$.

Let ${𝑆}_5$ be the symmetric group on $5$ letters. For each positive integer $𝑛$, list the number of elements of ${𝑆}_5$ of order $𝑛$. Justify your answer.

Let ${𝔽}_4$ be the field with $4$ elements. Let $𝐺=SL(2,{𝔽}_4)$ be the group of $2$ by $2$ invertible matrices with entries in ${𝔽}_4$. What is the order of $𝐺$? Show, by analysing the action of $𝐺$ on the lines containing the origin in ${({𝔽}_4)}^2$, that $𝐺$ is a simple group. (Hint: how many lines containing the origin are there?)

List, up to isomorphism, all commutative rings with $4$ elements. Prove your answer.