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…
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 $\sigma(g) = g$…