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## Prove that the augmentation ideal of a given group ring is nilpotent

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.29 Solution: We begin with some lemmas. Lemma 1: Let $\pi : G \rightarrow H$ be a…

## Classify groups of order 6

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.10 Solution: Let $G$ be a nonabelian group of order 6. We claim that if $x$ is…

## Every abelian simple group has prime order

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.4 Exercise 3.4.1 Solution: Let $G$ be an abelian simple group. Suppose $G$ is infinite. If $x \in G$…

## Normal subgroups whose order and index are coprime are unique up to order

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.19 Let $G$ be a finite group, $N \leq G$ a normal subgroup, and suppose that $|N|$…