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…
Finite subgroups with relatively prime orders intersect trivially
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.8 Let $G$ be a group and let $H, K \leq G$ be finite subgroups of relatively…
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…
If a group has a unique subgroup of a given order, then that subgroup is normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.5 Let $G$ be a group, $H$ a subgroup of $G$, and fix $g \in G$. (1)…
A finite group of width two has a trivial center
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.4 Let $G$ be a group. Prove that if $|G| = pq$ for some primes $p$ and…
Compute the subgroup lattice of quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.3 Compute the lattice of subgroups of $Q_8$. Solution: Every $\subseteq$-minimal subgroup of $Q_8$ is cyclic; disregarding…
The subgroup lattice of a symmetric group on three elements
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.2 Find the lattice of subgroups of $S_3$. Solution: Every (nontrivial) $\subseteq$-minimal subgroup of $S_3$ is cyclic…
Permissible subgroup orders in a group of order 120, and their indices
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.1 Which of the following are permissible orders for subgroups of a group of order 120: 1,…
Measure of open intervals and half-open half-closed intervals
Solution to Measure, Integration & Real Analysis by Axler Section 2A Exercise 2A6 Prove that if $a,b\in\mathbf R$ and $a<b$, then $$|(a,b)|=|(a,b]|=|[a,b)|=b-a.$$ Solution 1: Since $(a,b)\subset [a,b]$, by 2.5 and…