91Ó°ÊÓ

For each of the following groups \(G,\) determine whether \(H\) is a normal subgroup of \(G\). If \(H\) is a normal subgroup, write out a Cayley table for the factor group \(G / H\). (a) \(G=S_{4}\) and \(H=A_{4}\) (b) \(G=A_{5}\) and \(H=\\{(1),(123),(132)\\}\) (c) \(G=S_{4}\) and \(H=D_{4}\) (d) \(G=Q_{8}\) and \(H=\\{1,-1, I,-I\\}\) (e) \(G=\mathbb{Z}\) and \(H=5 \mathbb{Z}\)

Find all the subgroups of \(D_{4}\). Which subgroups are normal? What are all the factor groups of \(D_{4}\) up to isomorphism?

Find all the subgroups of the quaternion group, \(Q_{8}\). Which subgroups are normal? What are all the factor groups of \(Q_{8}\) up to isomorphism?

Show that the intersection of two normal subgroups is a normal subgroup.

If \(G\) is abelian, prove that \(G / H\) must also be abelian.

If \(G\) is cyclic, prove that \(G / H\) must also be cyclic.

Let \(H\) be a subgroup of index 2 of a group \(G\). Prove that \(H\) must be a normal subgroup of \(G\). Conclude that \(S_{n}\) is not simple for \(n \geq 3\).

Define the centralizer of an element \(g\) in a group \(G\) to be the set $$C(g)=\\{x \in G: x g=g x\\}$$ Show that \(C(g)\) is a subgroup of \(G\). If \(g\) generates a normal subgroup of \(G,\) prove that \(C(g)\) is normal in \(G\).

Recall that the center of a group \(G\) is the set $$Z(G)=\\{x \in G: x g=g x \text { for all } g \in G\\}$$ (a) Calculate the center of \(S_{3}\). (b) Calculate the center of \(G L_{2}(\mathbb{R})\). (c) Show that the center of any group \(G\) is a normal subgroup of \(G\). (d) If \(G / Z(G)\) is cyclic, show that \(G\) is abelian.

Let \(G\) be a group and let \(G^{\prime}=\left\langle a b a^{-1} b^{-1}\right\rangle ;\) that is, \(G^{\prime}\) is the subgroup of all finite products of elements in \(G\) of the form \(a b a^{-1} b^{-1}\). The subgroup \(G^{\prime}\) is called the commutator subgroup of \(G\). (a) Show that \(G^{\prime}\) is a normal subgroup of \(G\). (b) Let \(N\) be a normal subgroup of \(G\). Prove that \(G / N\) is abelian if and only if \(N\) contains the commutator subgroup of \(G\).