## 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…

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…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.2 Solution: Note that $gQg^{-1} \leq gHg^{-1}$. Moreover, because conjugation by g is an automorphism, $|gQg^{-1}| =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: If $P \leq H \leq G$ is a Sylow $p$-subgroup of $G$, then $p$ does…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.14 Solution: Let $p$ be the smallest prime dividing $n$, and write $n = pm$. Now $G$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.13 Solution: $G$ contains an element $x$ of order 2 by Cauchy’s Theorem. Let $\pi : G…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.12 Solution: If $\pi[G] \leq S_G$ contains an odd permutation, then $\pi[G] \not\leq A_G$. Now $A_G \vartriangleleft…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.11 Solution: This action of $G$ is faithful, so that the induced action of $H = \langle…

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…

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 $x \in G$…

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 $|N|$…