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)…
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…
Dih(24) and Sym(4) are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.9 Prove that $D_{24}$ and $S_4$ are not isomorphic. Solution: We know from Exercise 1.2.3 that $D_{24}$…
Finite symmetric groups of different orders are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.8 Prove that if $n \neq m$ then $S_n$ and $S_m$ are not isomorphic. Solution: We know…
Write out the Cayley table for Sym(3), Dih(8), and the quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.2 Write out the group tables for $S_3$, $D_8$, and $Q_8$. Solution: Group table for $S_3$:…