In a unital ring, (-1)² = 1
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.1 Let $R$ be a ring with 1. Show that $(-1)^2 = 1$ in $R$. Solution: Let…
Basic properties of Hall subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.10 Solution: We have $[HN:H \cap N] = [HN: H \cap N]$, so that $$[HN:N][N:H \cap N]…
Interaction of Sylow subgroups with quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.9 Let $p$ be a prime and let $G$ be a finite group of order $p^am$, where…
When the quotient by an intersection is isomorphic to a product of quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.7 Let $G$ be a group and let $M,N \leq G$ be normal such that $G =…
Compute the subgroup lattice of a quotient of modular group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.6 Let $M = \langle u,v \rangle$ be the modular group of order 16 described in Exercise…
A fact on the interaction of subgroups with subgroups of prime index
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.3 Let $G$ be a group, $N \leq G$ a normal subgroup of prime index $p$, and…
The Lattice Isomorphism Theorem
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.2 Prove all parts of the Lattice Isomorphism Theorem. Solution: Let $G$ be a group and $N…
The special linear group is normal in the general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.35 Let $F$ be a field and $n$ a positive integer. Prove that $SL_n(F)$ is normal in…
Compute the index of the special linear group in the general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.1 Let $F$ be a finite field of order $q$ and let $n$ be a positive integer.…
Perform explicit computation in a quotient of the modular group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.19 Let $G = M = \langle u,v \ |\ u^2 = v^8 = 1, vu =…