Isomorphism Problems

These problems are drawn from Chapter 9 of the text, page 121 and following.

  1. (Problem 2) Prove that $\mathbb{C}^{\times}$ is isomorphic to the subgroup of $\mathrm{GL}_{2}(\mathbb{R})$ consisting of matrices of the form \(\left(\begin{matrix} a & b \\ -b & a \end{matrix}\right).\) (Hint: let $a$ and $b$ be the real and imaginary parts of $z\in\mathbb{C}^{\times}$.)

  2. (Problem 12) Both $S_{4}$ and $D_{12}$ are non-abelian groups of order $24$. Prove that they are not isomorphic.

  3. (Problem 16) Find the order of the element $(6,15,4)$ in \(G=\mathbb{Z}_{30}\times\mathbb{Z}_{45}\times\mathbb{Z}_{24}.\)

  4. (Problem 24) Prove or disprove: There is a noncyclic abelian group of order $51$.

  5. (Problem 34) An automorphism of a group $G$ is an isomorphism $f:G\to G$. Prove that complex conjugation is an automorphism of the additive group of complex numbers; in other words, prove that $f(a+bi)=a-bi$ is an isomorphism from $\mathbb{C}\to\mathbb{C}$.

  6. (Problem 37) Let $G$ be a group and let $\mathrm{Aut}(G)$ be the set of automorphisms of $G$. Prove that $\mathrm{Aut}(G)$ is a group and is in fact a subgroup of the set of permutations of the elements of $G$.

  7. (Problem 38) Find the automorphism group $\mathrm{Aut}(\mathbb{Z}_{6})$.

  8. (Problem 52) Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively. Prove that $H_1\times H_2$ is a subgroup of $G_1\times G_2$.

BONUS PROBLEM: Problem 55 on page 124 of the text classifies all groups of order $2p$ where $p$ is a prime, showing that such a group is either cyclic of order $2p$ or is isomorphic to the dihedral group of symmetries of the regular $p$-gon $D_{p}$. It has 9 parts. It is worth doing!