## Given a subgroup of a group, the numbers of left and right cosets are equal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.12 Let $G$ be a group and $H \leq G$. Prove that the map $x \mapsto x^{-1}$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.12 Let $G$ be a group and $H \leq G$. Prove that the map $x \mapsto x^{-1}$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.11 Let $G$ be a group and $K \leq H \leq G$. Prove that $[G : K]…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.10 Let $G$ be a group and let $H,K \leq G$ be subgroups of finite index; say…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.8 Let $G$ be a group and let $H, K \leq G$ be finite subgroups of relatively…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.6 In $\mathbb{Z}/(48)$, write out all elements of $\langle \overline{a} \rangle$ for every $\overline{a}$. Find all inclusions…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.1 Find all subgroups of $G = \mathbb{Z}/(45)$, giving a generator for each. Describe the containments among…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.10 Let $G = \left\{ \left[{a \atop 0} {b \atop c}\right] \ |\ a,b,c \in \mathbb{R}, a…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.11 Let $G$ be a group. Prove that $Z(G) \leq N_G(A)$ for any subset $A \subseteq G$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.10 Let $H$ be a subgroup of order 2 in $G$. Show that $N_G(H) = C_G(H)$. Deduce…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.9 Let $G$ be a group, $H \leq G$, and $A \subseteq G$. Define $$N_H(A) = \{…