The intersection of a nonempty collection of subrings is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.4 Prove that the intersection of any nonempty collection of subrings of a ring is also a…
In a subring containing the identity, units are units in the ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.3 Let $R$ be a ring with identity and let $S \subseteq R$ be a subring containing…
In a unital ring, the negative of a unit is a unit
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.2 Let $R$ be a ring with 1. Prove that if $u$ is a unit in $R$,…
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…
The group of complex p-power roots of unity is a proper quotient of itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.8 Let $p$ be a prime and let $G$ be the group of $p-$power roots of 1 in…
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…
Draw subgroup lattice of a quotient of quasi-dihedral group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.5 Let $QD_{16} = \langle \sigma, \tau \rangle$ be the quasidihedral group of order 16 described in…