Characterization of the orbits of a group action as equivalence classes
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.18 Let $H$ be a group acting on a set $A$. Prove that the relation on $A$…
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 proof of Cauchy’s Theorem
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.9 Solution: (1) We prove this equality by attempting to choose an arbitrary element of $\mathcal{S}$. Note…
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…