## Exhibit quaternion group in Symmetric group via regular representation

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.4 Solution: Recall that $Q_8 = \langle i,j \rangle$. Now $i(1) = i$, $i(-1) = -i$, $i(i)…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.4 Solution: Recall that $Q_8 = \langle i,j \rangle$. Now $i(1) = i$, $i(-1) = -i$, $i(i)…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.2 Solution: We use the notation $\sigma(k) = \sigma \cdot k$. (1) $1 \mapsto 1$ (2) We…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.10 Solution: We saw previously (by counting) that $E$ has precisely $p+1$ distinct subgroups of order $p$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.9 Solution: We know that if $F$ is a finite field then $\mathsf{Aut}(F^n) \cong GL_n(F)$. This isomorphism…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.2 Solution: (1) Define a mapping $\varphi : G_I \rightarrow \times_{i \in I} G_i$ by $(\varphi(g))_i =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.9 Suppose $G \leq S_A$ acts transitively on $A$ and let $H \leq G$ be normal. Let…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.4 Let $G$ be a group acting on a set $A$. Show that the following sets are…

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 Solution: See Lemma 3 of Exercise 3.2.10.

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…