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…
Find a generating set for the augmentation ideal of a group ring
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]…
Characterize the center of a group ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.13 Let $\mathcal{K} = \{k_1, \ldots, k_m \}$ be a conjugacy class in the finite group $G$.…
Exhibit an element in the center of a group ring of finite group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.12 Let $R$ be a ring with $1 \neq 0$, and let $G = \{g_1, \ldots, g_n…
Compute in a group ring of symmetric group over Z/(3)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.11 Consider the following elements of the group ring $\mathbb{Z}/(3)[S_3]$: $$\alpha = 1(2\ 3) + 2(1\ 2\…
Compute in a group ring of symmetric group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.10 Consider the following elements of the integral group ring $\mathbb{Z}[S_3]$: $$\alpha = 3(1\ 2) - 5(2\…
Compute in a group ring over dihedral group of order 8
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.9 Let $\alpha = r + r^2 - 2s$ and $\beta = -3r^2 + rs$ be elements…