Set of invertible upper triangular matrices is a subgroup of general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.16 Let $n \in \mathbb{Z}^+$ and let $F$ be a field. Prove that the set $$UT_n(F) =…
The union of a chain of subgroups is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.15 Let $G$ be a group, and $\{ H_i \}_{i \in \mathbb{Z}}$ be an ascending chain of…
Set of all elements of a given order is not necessarily a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.14 Let $n \geq 3$. Show that $\{ x \in D_{2n} \ |\ x^2 = 1\}$ is…
Additive subgroups of the rationals which are closed under inversion are trivial
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.13 Let $H$ be a subgroup of the additive group of rational numbers with the property that…
The n-th powers and n-th roots of an abelian group are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.12 Let $A$ be an abelian group and fix $n \in \mathbb{Z}^+$. Prove that the following subsets…
Prove that given subsets of a direct product are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.11 Let $A$ and $B$ be groups. Prove that the following sets are subgroups of $A \times…
An arbitrary intersection of subgroups is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.10 Let $G$ be a group. (1) Prove that if $H$ and $K$ are subgroups of $G$,…
Special linear group is a subgroup of general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.9 Let $F$ be a field and define $SL_n(F) \subseteq GL_n(F)$ by $$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 $\mathbb{Z} \times \mathbb{Z}/(n)$.…