Perform computations in a quotient of dihedral group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.17 Let $G = D_{16}$ and let $H = \langle r^4 \rangle$. (1) Show that the order…
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…
Every normalizer contains the group center
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 $A \subseteq G$.…
Compute the center of Dih(2n)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.7 Let $n \in \mathbb{Z}$ with $n \geq 3$. Prove the following. (1) $Z(D_{2n}) = 1$ if…
Compute the centralizers of each element in Sym(3), Dih(8), and the quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.4 For each of the groups $S_3$, $D_8$, and $Q_8$, compute the centralizer of each element and…
The centralizer and normalizer of a group center is the group itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.2 Let $G$ be a group. Prove that $C_G(Z(G)) = G$ and deduce that $N_G(Z(G)) = G$.…
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