Chapter 2

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=A_{4} \quad S=\\{(123),(12)(34)\\} $$

Let \(H\) and \(K\) be subgroups of a finite group \(G\) with index \([G: H]=n\) and index \([G: K]=m .\) Show that \(\operatorname{lcm}(n, m) \leq[G: H \cap K] \leq n m .\)

Let \(H\) be a subgroup of the group \(G\). Show that the map \(a \rightarrow a^{1}\) determines a one-to-one, onto map between the left cosets of \(H\) and the right cosets of \(H\).

For any positive integer \(n\) show that \(n=\sum \phi(d)\), where the sum is taken over all positive divisors \(d\) of \(n\) and \(\phi\) is the Euler \(\phi\) -function.

In Exercises 35 through 38 determine whether the indicated map \(\phi\) is an isomorphism. Justify your answer. $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}, \text { where } \phi(n)=3 n $$

Show that the converse of Lagrange's theorem is false. (Hint: Show that \(A_{4}\) has no subgroup of order \(6 .)\)

Show that \(U(8)\) and \(U(12)\) are isomorphic.

Show that in \(C^{*}\) the subgroup \(\langle i\rangle\) generated by \(i\) is isomorphic to \(\mathbb{Z}_{4}\)

Show that \(\mathbb{Z}_{4}\) and the Klein 4 -group \(V\) of Example 1.1 .22 are not isomorphic.

Find four different subgroups of \(S_{4}\) that are isomorphic to \(S_{3}\).

Show that the alternating group \(A_{4}\) contains a subgroup isomorphic to the Klein 4-group \(V\).

Show that the dihedral group \(D_{4}\) contains a subgroup isomorphic to the Klein 4 group \(V\).