Right multiplication by the inverse is a left group action of a group on itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.15 Let $G$ be any group. Show that the mapping defined by $g \cdot a = ag^{-1}$…
Demonstrate that right multiplication is not a left group action of a group on itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.14 Let $G$ be a nonabelian group. Show that the action of $G$ on itself by $g…
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…
Exhibit Dih(8) as a subgroup of Sym(4)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.11 Write out the cycle decomposition of the eight permutations in $S_4$ corresponding to the elements of…
Determine when a given action is faithful
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.10 Let $k \in \mathbb{Z}^+$. (1) For which values of $k$ is the action of $S_A$ on…
Compute the action of two group elements on a set
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.9 Let $A$ be a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…
Sym(A) acts on the set of all subsets of A having some fixed cardinality
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.8 Let $A$ be a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…
The action of the nonzero elements of a field on a vector space by left multiplication forms a faithful group action
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.7 Let $F$ be a field and $V$ an $F$-vector space. The axioms for vector spaces over…
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…