Quotient of a product by a product is isomorphic to the product of quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.4 Let $A$ and $B$ be groups, with $C \leq A$ and $D \leq B$ normal. Prove…
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 =…
Perform explicit computation in a quotient of a quasi-dihedral group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.18 Let $G = QD_{16}$ be the quasidihedral group presented by $$\langle \sigma, \tau \ |\ \sigma^8…
Perform computations in a quotient of dihedral group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.17 Let $G = D_{16}$ and let $H = \langle r^4 \rangle$. (1) Show that the order…
Solution to Measure, Integration & Real Analysis by Axler
Chapter 1 Riemann Integration §1A Review: Riemann Integral §1B Riemann Integral Is Not Good Enough Chapter 2 Measures §2A Outer Measure on R (#1) (#2) (#3) (#4) (#5) (#6) §2B…
The quotient of a generating set is a generating set of the quotient
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.16 Let $G$ be a group and $N \leq G$ a normal subgroup. Show that if $G…