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…
Q/Z is isomorphic to the group of complex roots of unity
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.14 Consider the additive quotient group $\mathbb{Q}/\mathbb{Z}$. (1) Show that every coset of $\mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$ has…
Exhibit a group homomorphism on the Heisenberg group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.11 Let $F$ be a field and let $$G = \left\{ \begin{bmatrix} a & b \\ 0…
Exhibit the automorphisms of Z/(48)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.8 Let $Z_{48} = \langle x \rangle$. For which integers a does the map $\varphi_a$ defined by…
Find the subgroup lattice of the cyclic group of order 48
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.7 Let $Z_{48} = \langle x \rangle$ and use the isomorphism $\mathbb{Z}/(48) \cong Z_{48}$ given by $\overline{1}…
Compute the center of a Heisenberg group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.14 Let $F$ be a field and let $H(F)$ denote the Heisenberg group over $F$ as defined…
Direct product of groups is essentially associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.12 Let $A$, $B$, and $C$ be groups. Show that $A \times (B \times C) \cong (A…
Direct product of groups is essentially commutative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.11 Let $A$ and $B$ be groups. Prove that $A \times B \cong B \times A$. Solution:…
If A and B have the same cardinality, then Sym(A) and Sym(B) are isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.10 Let $\theta : \Delta \rightarrow \Omega$ be a bijection. Define $\varphi : S_\delta \rightarrow S_\Omega$ by…
Surjective group homomorphisms preserve abelianicity
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.3 If $\varphi : G \rightarrow H$ is an isomorphism, prove that $G$ is abelian if and…