Chapter 34

In the group \(Z_{24}\), let \(H=(4)\) and \(N=(6)\). a. List the elements in \(H N\) (which we might write \(H+N\) for these additive groups) and in \(H \cap N\). b. List the cosets in \(H N / N\), showing the elements in each coset. c. List the cosets in \(H /(H \cap N)\), showing the elements in each coset. d. Give the correspondence between \(H N / N\) and \(H /(H \cap N)\) described in the proof of Theorem 34.5.

Show directly from the definition of a normal subgroup that if \(H\) and \(N\) are subgroups of a group \(G\), and \(N\) is normal in \(G\), then \(H \cap N\) is normal in \(H\).

Theory 7\. Show directly from the definition of a normal subgroup that if \(H\) and \(N\) are subgroups of a group \(G\), and \(N\) is normal in \(G\), then \(H \cap N\) is normal in \(H\).

Show directly from the definition of a normal subgroup that if \(H\) and \(N\) are subgroups of a group \(G\), and \(N\) is normal in \(G\), then \(H \cap N\) is normal in \(H\).

Let \(H, K\), and \(L\) be normal subgroups of \(G\) with \(H

Let \(K\) and \(L\) be normal subgroups of \(G\) with \(K \vee L=G\), and \(K \cap L=(e)\). Show that \(G / K \simeq L\) and \(G / L \simeq K\).