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Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.6 Solution: We compute the permutation representation of $G$ in $D_{G/N}$.$$1 \mapsto 1$$ $$r(N) = rN, r(rN)… Compute permutation representations of Dihedral group of order 8 Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.5 Solution: To save effort, we compute the representation of D_8 in S_{D_8}.$$1 \mapsto 1r(H) =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.10 Let $G$ be a group and $H,K \leq G$ subgroups. For each $x \in G$, define…

Given a subgroup of a group, the numbers of left and right cosets are equal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.12 Let $G$ be a group and $H \leq G$. Prove that the map $x \mapsto x^{-1}$…

Alternate characterization of cosets as equivalence classes

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.7 Let $G$ be a group and $H \leq G$. Define a relation $\sim$ on $G$ by…

Consequences of equality among left and right cosets

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.6 Let $G$ be a group, $H \leq G$ a subgroup, and $g \in G$ a fixed…

Compute the order of a quotient group element

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.5 Let $G$ be a group and $N$ a normal subgroup of $G$. Prove that the order…