Prove that a given function is a ring homomorphism and describe its kernel
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.7 Solution: Let $A,B \in R$ be arbitrary, with $A = \begin{bmatrix} a_1 & b_1 \\ 0…
Computation of all ring homomorphisms from Z to Z/(30)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.4 Find all ring homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}/(30)$. In each case describe the kernel and the…
Generalized coordinate subgroups of a direct product
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.2 Solution: (1) Define a mapping $\varphi : G_I \rightarrow \times_{i \in I} G_i$ by $(\varphi(g))_i =…
Stabilizer commutes with conjugation II
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.2 Let $G$ be a permutation group on the set $A$ (i.e. $G \leq S_A$), let $\sigma…
Stabilizer commutes with conjugation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.1 Solution: First we prove that $\mathsf{stab}_G(b) = g \mathsf{stab}_G(a) g^{-1}$. ($\subseteq$) If $x \in \mathsf{stab}_G(b)$, then…
Compute the kernel of the left regular action of a group on itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.13 Find the kernel of the left regular action. Solution: Recall that the left regular action of…
A group action is faithful precisely when the kernel of the corresponding permutation representation is trivial
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.6 Prove that a group $G$ acts faithfully on a set $A$ if and only if the…
The kernel of a group action is precisely the kernel of the induced permutation representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.5 Prove that the kernel of an action of the group $G$ on a set $A$ is…
The kernel and stabilizers of a group action are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.4 Let $G$ be a group acting on a set $A$. Show that the following sets are…
Exhibit a group homomorphism on the Heisenberg group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.11 Let $F$ be a field and let $$G = \left\{ \begin{bmatrix} a & b \\ 0…
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