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 3.1 Exercise 3.1.16 Let $G$ be a group and $N \leq G$ a normal subgroup. Show that if $G…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.8 Let $Z_{48} = \langle x \rangle$. For which integers a does the map $\varphi_a$ defined by…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.5 Find the number of generators for $\mathbb{Z}/(49000)$. Solution: The number of generators of $\mathbb{Z}/(n)$ is $\varphi(n)$,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.4 Find all generators for $\mathbb{Z}/(202)$. Solution: The generators of $\mathbb{Z}/(202)$ are precisely those (equivalence classes represented…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.3 Find all the generators in $\mathbb{Z}/(48)$. Solution: The generators of $\mathbb{Z}/(48)$ are precisely those (equivalence classes…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.24 Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order…