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## Bounds on the index of an intersection of two subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.10 Let $G$ be a group and let $H,K \leq G$ be subgroups of finite index; say…

## 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…