Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.4 Solution: ($\Rightarrow$) Suppose $R$ is a field. Let $I \subseteq R$ be an ideal which properly…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.2 Solution: Recall that the augmentation ideal of $R[G]$ is the kernel of the ring homomorphism $R[G]…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.1 Solution: In Exercise 7.2.6, we saw that $AE_{i,j}$ is the matrix whose j-th column is the…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.8 Solution: $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, so that the Sylow 2-subgroups of…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.7 Solution: The Sylow 2-subgroups of $S_4$ have order 8, and by Sylow’s Theorem, $n_2 = 1…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.5 Solution: Let $P \leq D_{2n}$ be a Sylow $p$-subgroup, where $p$ is an odd prime. Note…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: The Sylow 2-subgroups of $D_{12}$ have order 4. By Sylow’s Theorem, the number $n_2$ of…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: Let $G$ be a finite group and $p$ a prime dividing $|G|$. By Sylow’s Theorem,…
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…