Linearity, Author at Solutions to Linear Algebra Done Right

Additive subgroups of the rationals which are closed under inversion are trivial

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.13 Let $H$ be a subgroup of the additive group of rational numbers with the property that…

The n-th powers and n-th roots of an abelian group are subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.12 Let $A$ be an abelian group and fix $n \in \mathbb{Z}^+$. Prove that the following subsets…

Prove that given subsets of a direct product are subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.11 Let $A$ and $B$ be groups. Prove that the following sets are subgroups of $A \times…

An arbitrary intersection of subgroups is a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.10 Let $G$ be a group. (1) Prove that if $H$ and $K$ are subgroups of $G$,…

Special linear group is a subgroup of general linear group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.9 Let $F$ be a field and define $SL_n(F) \subseteq GL_n(F)$ by $$SL_n(F) = \{ A \in…

Union of two subgroups is a subgroup if and only if one is contained in the other

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.8 Let $H$ and $K$ be subgroups of $G$. Prove that $H \cup K$ is a subgroup…

Compute a torsion subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.7 Fix $n \in \mathbb{Z}^+$ with $n > 1$. Find the torsion subgroup of $\mathbb{Z} \times \mathbb{Z}/(n)$.…

Torsion elements in an abelian group form a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.6 Let $G$ be a group. An element $x \in G$ is called torsion if it has…

Finite groups with at least 3 elements cannot have a subgroup consisting of all but one element

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.5 Let $G$ be a finite group with $|G| = n > 2$. Prove that $G$ cannot…

Exhibit an infinite subset in a group which is multiplicatively closed but not a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.4 Give an explicit example of a group $G$ and an infinite subset $H$ of $G$ which…