Chapter 15

The center of a group \(G\) is the normal subgroup \(C\) of \(G\) consisting of all those elements of \(G\) which commute with every element of \(G\). Suppose the quotient group \(G / C\) is a cyclic group; say it is generated by the element \(C a\) of \(G / C\). Prove each of the following: For every \(x \in G\), there is some integer \(m\) such that \(C x=C a^{m}\).

There are some group properties which, if they are true in \(G / H\) and in \(H\), must be true in \(G .\) Here is a sampling. Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove: If every element of \(G / H\) has finite order, and every element of \(H\) has finite order, then every element of \(G\) has finite order.

In each of the following, \(H\) is a subset of \(\mathbb{R} \times \mathbb{R}\) (a) Prove that \(H\) is a normal subgroup of \(\mathbb{R} \times \mathbb{R}\). (Remember that every subgroup of an abelian group is normal.) (b) In geometrical terms, describe the elements of the quotient group \(G / H\). (c) In geometrical terms or otherwise, describe the operation of \(G / H\). $$ H=\\{(x, 0): x \in \mathbb{R}\\} $$

In each of the following, \(G\) is a group and \(H\) is a normal subgroup of \(G\). List the elements of \(G / H\) and then write the table of \(G / H\). Example \(G=\mathbb{Z}_{6} \quad\) and \(\quad H=\\{0,3\\}\) The elements of \(G / H\) are the three cosets \(H=H+0=\\{0,3\\}, H+1=\\{1,4\\}\), and \(H+2=\\{2,5\\} .\) (Note that \(H+3\) is the same as \(H+0, H+4\) is the same as \(H+1\), and \(H+5\) is the same as \(H+2\).) The table of \(G / H\) is $$ \begin{array}{c|ccc} \+ & H & H+1 & H+2 \\ \hline H & H & H+1 & H+2 \\ H+1 & H+1 & H+2 & H \\ H+2 & H+2 & H & H+1 \end{array} $$ \(G=\mathbb{Z}_{10}, H=\\{0,5\\} .\left(\right.\) Explain why \(\left.G / H \cong \mathbb{Z}_{5} .\right)\)

The center of a group \(G\) is the normal subgroup \(C\) of \(G\) consisting of all those elements of \(G\) which commute with every element of \(G\). Suppose the quotient group \(G / C\) is a cyclic group; say it is generated by the element \(C a\) of \(G / C\). Prove each of the following: For every \(x \in G\), there is some integer \(m\) such that \(x=c a^{m}\), where \(c \in C\).

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: If \((G: H)=m\), the order of every element of \(G / H\) is a divisor of \(m\).

There are some group properties which, if they are true in \(G / H\) and in \(H\), must be true in \(G .\) Here is a sampling. Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove: If every element of \(G / H\) has a square root, and every element of \(H\) has a square root, then every element of \(G\) has a square root.

In each of the following, \(H\) is a subset of \(\mathbb{R} \times \mathbb{R}\) (a) Prove that \(H\) is a normal subgroup of \(\mathbb{R} \times \mathbb{R}\). (Remember that every subgroup of an abelian group is normal.) (b) In geometrical terms, describe the elements of the quotient group \(G / H\). (c) In geometrical terms or otherwise, describe the operation of \(G / H\). $$ H=\\{(x, y): y=-x\\} $$

In each of the following, \(G\) is a group and \(H\) is a normal subgroup of \(G\). List the elements of \(G / H\) and then write the table of \(G / H\). Example \(G=\mathbb{Z}_{6} \quad\) and \(\quad H=\\{0,3\\}\) The elements of \(G / H\) are the three cosets \(H=H+0=\\{0,3\\}, H+1=\\{1,4\\}\), and \(H+2=\\{2,5\\} .\) (Note that \(H+3\) is the same as \(H+0, H+4\) is the same as \(H+1\), and \(H+5\) is the same as \(H+2\).) The table of \(G / H\) is $$ \begin{array}{c|ccc} \+ & H & H+1 & H+2 \\ \hline H & H & H+1 & H+2 \\ H+1 & H+1 & H+2 & H \\ H+2 & H+2 & H & H+1 \end{array} $$ \(G=S_{3}, H=\\{\varepsilon, \beta, \delta\\}\).

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: If \((G: H)=p\), where \(p\) is a prime, then the order of every element \(a \notin H\) in \(G\) is a multiple of \(p\). [Use (1).]

There are some group properties which, if they are true in \(G / H\) and in \(H\), must be true in \(G .\) Here is a sampling. Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove: Let \(p\) be a prime number. A group \(G\) is called a \(p\)-group if the order of every element \(x\) in \(G\) is a power of \(p .\) Prove: If \(G / H\) and \(H\) are \(p\)-groups, then \(G\) is a \(p\)-group.

The center of a group \(G\) is the normal subgroup \(C\) of \(G\) consisting of all those elements of \(G\) which commute with every element of \(G\). Suppose the quotient group \(G / C\) is a cyclic group; say it is generated by the element \(C a\) of \(G / C\). Prove each of the following: Conclude that if \(G / C\) is cyclic, then \(G\) is abelian.

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: If \(G\) has a normal subgroup of index \(p\), where \(p\) is a prime, then \(G\) has at least one element of order \(p\).

There are some group properties which, if they are true in \(G / H\) and in \(H\), must be true in \(G .\) Here is a sampling. Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove: If \(G / H\) and \(H\) are finitely generated, then \(G\) is finitely generated. (A group is said to be finitely generated if it is generated by a finite subset of its elements.)

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: If \((G: H)=m\), then \(a^{m} \in H\) for every \(a \in G\).

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: In \(\mathbb{Q} / \mathbb{Z}\), every element has finite order.

In each of the following, \(G\) is a group and \(H\) is a normal subgroup of \(G\). List the elements of \(G / H\) and then write the table of \(G / H\). Example \(G=\mathbb{Z}_{6} \quad\) and \(\quad H=\\{0,3\\}\) The elements of \(G / H\) are the three cosets \(H=H+0=\\{0,3\\}, H+1=\\{1,4\\}\), and \(H+2=\\{2,5\\} .\) (Note that \(H+3\) is the same as \(H+0, H+4\) is the same as \(H+1\), and \(H+5\) is the same as \(H+2\).) The table of \(G / H\) is $$ \begin{array}{c|ccc} \+ & H & H+1 & H+2 \\ \hline H & H & H+1 & H+2 \\ H+1 & H+1 & H+2 & H \\ H+2 & H+2 & H & H+1 \end{array} $$ \(G=P_{3}, H=\\{\emptyset,\\{1\\}\\} .\left(P_{3}\right.\) is the group of subsets of \(\left.\\{1,2,3\\} .\right)\)