Demonstrate that given subsets of Dih(8) are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.3 Show that the following subsets of $D_8$ are subgroups. (1) $\{ 1, r^2, s, sr^2 \}$…
Demonstrate that a given subset is not a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.2 Show that in each of the following examples, the specified subset is not a subgroup. (1)…
Decide whether or not a given subset of a group is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.1 For each of the following, show that the specified subset is a subgroup of the given…
Identify quaternion group as a subgroup of the general linear group of dimension 2 over complex numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.26 Let $i$ and $j$ be the generators of $Q_8 = \langle i, j \ |\ i^4…
Identify Dih(2n) as a subgroup of general linear group of dimension 2 over real numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.25 Let $n \in \mathbb{Z}^+$, let $r$ and $s$ be the usual generators of $D_{2n}$, and let…
If a finite group has a generating set containing two elements of order 2, then it is a dihedral group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.24 Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order…
If a group has an automorphism which is fixed point free of order 2, then it is abelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.23 Let $G$ be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g) = g$…
Power maps are abelian group homomorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.22 Let $A$ be an abelian group and fix some $k \in \mathbb{Z}$. Prove that the map…
Exhibit some automorphisms of the rational numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.21 Prove that for each fixed nonzero $k \in \mathbb{Q}$ the map $\varphi : \mathbb{Q} \rightarrow \mathbb{Q}$…
The set of all group automorphisms of a fixed group is a group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.20 Let $G$ be a group and let $\mathsf{Aut}(G)$ be the set of all isomorphisms $G \rightarrow…