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Give an example of a semidirect product of two cyclic groups of odd order that is not abelian.

Let $𝐺$ be finite, let $𝑝$ divide $|𝐺|$, and let ${𝑂}_{𝑝}$ be the intersection of all Sylow $𝑝$-subgroups of $𝐺$. Prove that ${𝑂}_{𝑝}$ is normal in $𝐺$ and contains every normal $𝑝$-subgroup of $𝐺$.

Let $𝑅$ be a commutative ring with identity and let $𝐼\text{◃}𝑅$ satisfy $𝑅/𝐼\cong \mathbb{Z}[𝑖]$.

(a) Prove that if $𝐽$ is a maximal ideal containing $𝐼$, then $𝑅/𝐽$ is finite.

(b) Find distinct maximal ideals ${𝐽}_0,{𝐽}_1$ containing $𝐼$ such that $𝑅/{𝐽}_0\cong 𝑅/{𝐽}_1$.

Let $𝑅$ be a commutative ring, let $𝑓\in 𝑅$, and let ${𝑅}_{𝑓}={𝑆}^{-1}𝑅$ for $𝑆=\{1,𝑓,{𝑓}^2,\dots \}$. Prove that

Let $𝑀$ be a module over a commutative ring $𝑅$, and let $𝑁\subseteq 𝑀$ be a submodule. If $𝑁$ and $𝑀/𝑁$ are finitely generated, prove that $𝑀$ is finitely generated.

Let $𝐴$ be a matrix over ${𝔽}_3$ satisfying

Prove that $𝐴$ is diagonalizable over ${𝔽}_{81}$.

Let $𝐾/𝐹$ be Galois with $\text{Gal}(𝐾/𝐹)\cong {𝑆}_3$.

(a) Describe the diagram of intermediate fields and label the degrees.

(b) How many intermediate fields $𝐸$, with $𝐹\subseteq 𝐸\subseteq 𝐾$, are Galois over $𝐹$?

Let $𝐹$ have characteristic $𝑝>0$, and suppose

has no root in $𝐹$. Prove that $𝑓$ is irreducible over $𝐹$.

(a) Suppose $𝑅$ is a discrete valuation ring, with residue field $𝑘$. Give a projective resolution of $𝑘$ over $𝑅$ and compute ${Ext}_{𝑅}^{𝑖}(𝑘,𝑘)$ for all $𝑖$ (where the action of $𝑅$ on $𝑘$ is the action of $𝑅$ on its residue field).

(b) Suppose $𝑘$ is a field, and let $𝑅=𝑘[𝑥,𝑦]$. Let $𝔪=(𝑥-1,𝑦)$ and let $𝑀=𝑅/𝔪$. Thus $𝑀$ is an $𝑅$-module that is one-dimensional as a $𝑘$-vector space. Give a projective resolution of $𝑀$ and compute ${Ext}_{𝑅}^{𝑖}(𝑀,𝑀)$ for all $𝑖$.

A finite group $𝐺$ has character table:

(a) If ${𝜁}_1,\dots ,{𝜁}_{𝑛}$ are roots of unity with ${𝜁}_1+\dots +{𝜁}_{𝑛}=𝑛$ show that each ${𝜁}_{𝑖}$ is 1.

(b) Prove that $𝑁={𝐶}_1\cup {𝐶}_2$ is a normal subgroup of $𝐺$.

(c) Use the character table to compute $|𝐺|$ and $|{𝐶}_2|$, and thus determine the order of $𝑁$.

Note: This problem is designed to be done without identifying the group $𝐺$, illustrating how the character table of any group contains enough information to characterize its normal subgroups and other important information. So if you happen to know a group that has this character table your answer may only make use of that identification if you prove that yours is the only group with this character table.