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…
In a finite group, conjugation permutes sub-Sylow subgroups
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}| =…
Sylow subgroups of a finite group which are contained in some other subgroup are also Sylow there
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…
A finite group of composite order n having a subgroup of every order dividing n is not simple
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 $G$…
If a group has order 2k where k is odd, then it has a subgroup of index 2
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…
If regular representation of a group G contains an odd permutation, then G has a subgroup of index 2
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…
Characterization of parity in the left regular representation of a finite group
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…
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 $|N|$…