Chapter 5

Let \(G\) be the group \(\left\\{e, a, b, b^{2}, a b, a b^{2}\right\\}\) whose generators satisfy: \(a^{2}=e, b^{3}=e\), \(b a=a b^{2}\). Write the table of \(G\).

List all the cyclic subgroups of \(\left\langle\mathbb{Z}_{10},+\right\rangle\).

If \(H\) and \(K\) are subgroups of a group \(G\), prove that \(H \cap K\) is a subgroup of \(G\). (Remember that \(x \in H \cap K \quad\) iff \(\quad x \in H \quad\) and \(\quad x \in K)\).

Let \(G\) be an abelian group. If \(H=\left\\{x \in G: x=x^{-1}\right\\}\), that is, \(H\) consists of all the elements of \(G\) which are their own inverses, prove that \(H\) is a subgroup of \(G\).

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle F(\mathbb{F}),+\rangle, H=\\{f \in \mathscr{F}(\mathbb{R}): f(x)=0\) for every \(x \in[0,1]\\}\)

In each of the following, determine whether or not \(H\) is a subgroup of \(G\). (Assume that the operation of \(H\) is the same as that of \(G\).) \(G=\langle R,+\rangle, H=\\{\log a: a \in \mathbb{Q}, a>0\\} . \quad H\) is \(\square \quad\) is not \(\square\) a subgroup of \(G\)

Let \(G\) be the group \(\left\\{e, a, b, b^{2}, b^{3}, a b, a b^{2}, a b^{3}\right\\}\) whose generators satisfy: \(a^{2}=e\), \(b^{4}=e, b a=a b^{3} .\) Write the table of \(G .\left(G\right.\) is called the dihedral group \(D_{4}\).)

Show that \(\mathbb{Z}_{10}\) is generated by 2 and 5 .

Let \(H\) and \(K\) be subgroups of \(G\). If \(H \subseteq K\), then \(H\) is a subgroup of \(K\).

Let \(G\) be an abelian group. Let \(n\) be a fixed integer, and let \(H=\left\\{x \in G: x^{n}=e\right\\}\). Prove that \(H\) is a subgroup of \(G\).

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle\mathscr{F}(\mathbb{R}),+\rangle, H=\\{f \in \mathscr{F}(\mathbb{R}): f(-x)=-f(x)\\}\)

In each of the following, determine whether or not \(H\) is a subgroup of \(G\). (Assume that the operation of \(H\) is the same as that of \(G\).) \(G=\langle\mathbf{R},+\rangle, H=\\{\log n: n \in \mathbb{Z}, n>0\\} . \quad H\) is \(\square\) is not \(\square\) a subgroup of \(G\)

Describe the subgroup of \(\mathbb{Z}_{12}\) generated by 6 and \(9 .\)

By the center of a group \(G\) we mean the set of all the elements of \(G\) which commute with every element of \(G\), that is, $$ C=\\{a \in G: a x=x a \text { for every } x \in G\\} $$ Prove that \(C\) is a subgroup of \(G\). (HINT: If we wish to assume \(x y=y x\) and prove \(x y^{-1}=y^{-1} x\), it is best to prove first that \(y x y^{-1}=x\).)

Let \(G\) be an abelian group. Let \(H=\left\\{x \in G: x=y^{2}\right.\) for some \(\left.y \in G\right\\}\), that is, let \(H\) be the set of all the elements of \(G\) which have a square root. Prove that \(H\) is a subgroup of \(G\).

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle\mathscr{F}(\mathbb{R}),+\rangle, H=\\{f \in \mathscr{F}(R): f\) is periodic of period \(\pi\\}\)

In each of the following, determine whether or not \(H\) is a subgroup of \(G\). (Assume that the operation of \(H\) is the same as that of \(G\).) \(G=\langle R,+\rangle, H=\\{x \in \mathbb{R}: \tan x \in \mathbb{Q}\\}, \quad H\) is \(\square\) is not \(\square\) a subgroup of \(G\). HinT: Use the following formula from trigonometry: $$ \tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y} $$

Let \(G\) be the commutative group \(\\{e, a, b, c, a b, b c, a c, a b c\\}\) whose generators satisfy: \(a^{2}=b^{2}=c^{2}=e\). Write the table of \(G\).

Describe the subgroup of \(\mathbb{Z}\) generated by 10 and 15 .

Let \(C^{\prime}=\left\\{a \in G:(a x)^{2}=(x a)^{2}\right.\) for every \(\left.x \in G\right\\}\). Prove that \(C^{\prime}\) is a subgroup of \(G\).Let \(C^{\prime}=\left\\{a \in G:(a x)^{2}=(x a)^{2}\right.\) for every \(\left.x \in G\right\\}\). Prove that \(C^{\prime}\) is a subgroup of \(G\).

Let \(G\) be an abelian group. Let \(H\) be a subgroup of \(G\), and let \(K=\left\\{x \in G: x^{2} \in H\right\\}\). Prove that \(K\) is a subgroup of \(G\).

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle\delta(\mathbb{R}),+\rangle, H=\left\\{f \in^{\prime} G(\mathbb{R}): \int_{0}^{1} f(x) d x=0\right\\}\)

In each of the following, determine whether or not \(H\) is a subgroup of \(G\). (Assume that the operation of \(H\) is the same as that of \(G\).) \(G=\left\langle\mathbb{R}^{*}, \cdot\right\rangle, H=\left\\{2^{n} 3^{m}: m, n \in \mathbb{Z}\right\\} . \quad H\) is \(\square \quad\) is not \(\square \quad\) a subgroup of \(G\)

Let \(G\) be a finite group, say a group with \(n\) elements, and let \(S\) be a nonempty subset of \(G\). Suppose \(e \in S\), and that \(S\) is closed with respect to multiplication. Prove that \(S\) is a subgroup of \(G\). (HINT: It remains to prove that \(G\) is closed with respect to inverses. Let \(G=\left\\{a_{1}, \ldots, a_{n}\right\\} ;\) one of these elements is \(e .\) If \(a_{i} \in G\), consider the distinct elements \(\left.a_{i} a_{1}, a_{i} a_{2}, \ldots, a_{i} a_{n},\right\),

Let \(G\) be an abelian group. Let \(H\) be a subgroup of \(G\), and let \(K\) consist of all the elements \(x\) in \(G\) such that some power of \(x\) is in \(H\). That is, \(K=\left\\{x \in G:\right.\) for some integer \(\left.n>0, x^{n} \in H\right\\}\). Prove that \(K\) is a subgroup of \(G\).

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle\mathscr{I}(\mathbb{R}),+\rangle, H=\\{f \in \mathscr{S}(\mathbb{R}): d f / d x\) is constant \(\\}\)

In each of the following, determine whether or not \(H\) is a subgroup of \(G\). (Assume that the operation of \(H\) is the same as that of \(G\).) \(G=\langle\mathbb{R} \times \mathbb{R},+\rangle, H=\\{(x, y): y=2 x\\} . \quad H\) is \(\square\) is not \(\square\) a subgroup of \(G\)

Show that \(\mathbb{Z}_{2} \times \mathbb{Z}_{3}\) is a cyclic group. Show that \(\mathbb{Z}_{3} \times \mathbb{Z}_{4}\) is a cyclic group.

Let \(G\) be a group, and \(f: G \rightarrow G\) a function. A period of \(f\) is any element \(a\) in \(G\) such that \(f(x)=f(a x)\) for every \(x \in G\). Prove: The set of all the periods of \(f\) is a subgroup of \(G\).

Let \(G\) be an abelian group. Suppose \(H\) and \(K\) are subgroups of \(G\), and define \(H K\) as follows: $$ H K=\\{x y: x \in H \quad \text { and } y \in K\\} $$ Prove that \(H K\) is a subgroup of \(G\).

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle F(\mathbb{R}),+\rangle, H=\\{f \in \mathscr{F}(R): f(x) \in \mathbb{Z}\) for every \(x \in \mathbb{R}\\}\)

Show that \(\mathbb{Z}_{2} \times \mathbb{Z}_{4}\) is not a cyclic group, but is generated by \((1,1)\) and \((1,2)\).

Let \(H\) be a subgroup of \(G\), and let \(K=\left\\{x \cdot \in G: x a x^{-1} \in H\right.\) for every \(\left.a \in H\right\\}\). Prove: (a) \(K\) is a subgroup of \(G\); (b) \(H\) is a subgroup of \(K\).

Suppose a group \(G\) is generated by two elements \(a\) and \(b\). If \(a b=b a\), prove that \(G\) is abelian.

Let \(G\) and \(H\) be groups, and \(G \times H\) their direct product. (a) Prove that \(\\{(x, e): x \in G\\}\) is a subgroup of \(G \times H\). (b) Prove that \(\\{(x, x): x \in G\\}\) is a subgroup of \(G \times G\).