Chapter 16

As a provisional definition, let us call a finite abelian group "decomposable" if there are elements \(a_{1}, \ldots, a_{n} \in G\) such that: (D1) For every \(x \in G\), there are integers \(k_{1}, \ldots, k_{n}\) such that \(x=a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{n}^{k_{n}}\). (D2) If there are integers \(l_{1}, \ldots, l_{n}\) such that \(a_{1}^{l_{1}} a_{2}^{l_{2}} \cdots a_{n}^{l_{n}}=e\) then \(a_{1}^{l_{1}}=a_{2}^{t_{2}}=\cdots\) \(=a_{n}^{l_{n}}=e\) If (D1) and (D2) hold, we will write \(G=\left[a_{1}, a_{2}, \ldots, a_{n}\right]\). Let \(G^{\prime}\) be the set of all products \(a_{2}^{l_{2}} \cdots a_{n}^{l_{8}}\), as \(l_{2}, \ldots, l_{n}\) range over \(\mathbb{Z}\). Prove that \(G^{\prime}\) is a subgroup of \(G\), and \(G^{\prime}=\left[a_{2}, \ldots, a_{n}\right]\).

Let \(G\) be a finite group, and \(K\) a \(p\)-Sylow subgroup of \(G\). Let \(X\) be the set of all the conjugates of \(K\). See Exercise M2. If \(C_{1}, C_{2} \in X\), let \(C_{1} \sim C_{2}\) iff \(C_{1}=a C_{2} a^{-1}\) for some \(a \in K\). Prove that \(\sim\) is an equivalence relation on \(X\).

If \(G\) is a group and \(p\) is any prime divisor of \(|G|\), it will be shown here that \(G\) has at least one element of order \(p\). This has already been shown for abelian groups in Chapter 15 , Exercise \(\mathrm{H} 4\). Thus, assume here that \(G\) is not abelian. The argument will proceed by induction; thus, let \(|G|=k\), and assume our claim is true for any group of order less than \(k\). Let \(\mathbf{C}\) be the center of \(G\), let \(C_{a}\) be the centralizer of \(a\) for each \(a \in G\), and let \(k=c+k_{s}+\cdots+k_{t}\) be the class equation of \(G\), as in Chapter 15, Exercise G2. Prove : If \(p\) is a factor of \(\left|C_{a}\right|\) for any \(a \in G\), where \(a \notin \mathbf{C}\), we are done. (Explain why.)

Let \(f\) be a homomorphism from \(G\) onto \(H\) with kernel \(K\) : $$ G \underset{K}{R} H $$ If \(S\) is any subgroup of \(H\), let \(S^{*}=\\{x \in G: f(x) \in S\\}\). Prove: $$ S^{*} \text { is a subgroup of } G \text {. } $$

Let \(H\) and \(K\) be normal subgroups of a group \(G\), with \(H \subseteq K\). Define \(\phi: G / H \rightarrow G / K\) by: \(\phi(H a)=K a\). Prove the following : $$ \phi \text { is a well-defined function. [That is, if } H a=H b \text {, then } \phi(H a)=\phi(H b) \text {. }] $$

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). For each \(x \in \mathbb{R}\), it is conventional to write cis \(x=\cos x+i \sin x\). Prove that cis \((x+y)=(\operatorname{cis} x)(\) cis \(y)\).

If \(H\) is a subgroup of a group \(G\), let \(X\) designate the set of all the cosets of \(H\) in \(G\). For each element \(a \in G\), define \(\rho_{a}: X \rightarrow X\) as follows: $$ \rho_{a}(H x)=H(x a) $$ Prove that each \(\rho_{a}\) is a permutation of \(X\).

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : $$ H \cap K \text { is a normal subgroup of } K \text {. } $$

Let \(G\) and \(H\) be groups. Suppose \(J\) is a normal subgroup of \(G\) and \(K\) is a normal subgroup of \(H\) Show that the function \(f(x, y)=(J x, K y)\) is a homomorphism from \(G \times H\) onto \((G / J) \times(H / K)\)

The symbol Aut \((G)\) is used to designate the set of all the automorphisms of \(G\). Prove that the set Aut \((G)\), with the operation o of composition, is a group by proving that \(\operatorname{Aut}(G)\) is a subgroup of \(S_{G}\).

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

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. $$ \mathbb{Z}_{5} \text { and } \mathbb{Z}_{20} /\langle 5\rangle $$

As a provisional definition, let us call a finite abelian group "decomposable" if there are elements \(a_{1}, \ldots, a_{n} \in G\) such that: (D1) For every \(x \in G\), there are integers \(k_{1}, \ldots, k_{n}\) such that \(x=a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{n}^{k_{n}}\). (D2) If there are integers \(l_{1}, \ldots, l_{n}\) such that \(a_{1}^{l_{1}} a_{2}^{l_{2}} \cdots a_{n}^{l_{n}}=e\) then \(a_{1}^{l_{1}}=a_{2}^{t_{2}}=\cdots\) \(=a_{n}^{l_{n}}=e\) If (D1) and (D2) hold, we will write \(G=\left[a_{1}, a_{2}, \ldots, a_{n}\right]\). Prove: \(G \cong\left\langle a_{1}\right\rangle \times G^{\prime}\). Conclude that \(G \cong\left\langle a_{1}\right\rangle \times\left\langle a_{2}\right\rangle \times \cdots \times\left\langle a_{n}\right\rangle .\)

The purpose of this exercise is to prove a property of cosets which is needed in Exercise Q. Let \(G\) be a finite abelian group, and let \(a\) be an element of \(G\) such that \(\operatorname{ord}(a)\) is a multiple of \(\operatorname{ord}(x)\) for every \(x \in G\). Let \(H=\langle a\rangle\). We will prove: For every \(x \in G\), there is some \(y \in G\) such that \(H x=H y\) and \(\operatorname{ord}(y)=\operatorname{ord}(H y)\). This means that every coset of \(H\) contains an element \(y\) whose order is the same as the coset's order. Let \(x\) be any element in \(G\), and let \(\operatorname{ord}(a)=t, \operatorname{ord}(x)=s\), and \(\operatorname{ord}(H x)=r\). Deduce from our hypotheses that \(r\) divides \(s\), and \(s\) divides \(t\). Thus, we may write \(s=r u\) and \(t=s v\), so in particular, \(t=r u v\).

Let \(p\) be a prime number. A finite group \(G\) is called a \(p\)-group if the order of every element \(x\) in \(G\) is a power \(p\). (The orders of different elements may be different powers of \(p\).) If \(H\) is a subgroup of any finite group \(G\), and \(H\) is a \(p\)-group, we call \(H\) a \(p\)-subgroup of \(G .\) Finally, if \(K\) is a \(p\)-subgroup of \(G\), and \(K\) is maximal (in the sense that \(K\) is not contained in any larger \(p\)-subgroup of \(G\) ), then \(K\) is called a \(p\)-Sylow subgroup of \(G\). Prove the following: Every conjugate of a \(p\)-Sylow subgroup of \(G\) is a \(p\)-Sylow subgroup of \(G\).

Let \(f\) be a homomorphism from \(G\) onto \(H\) with kernel \(K\) : $$ G \underset{K}{R} H $$ If \(S\) is any subgroup of \(H\), let \(S^{*}=\\{x \in G: f(x) \in S\\}\). Prove: $$ K \subseteq S^{*} $$

Let \(H\) and \(K\) be normal subgroups of a group \(G\), with \(H \subseteq K\). Define \(\phi: G / H \rightarrow G / K\) by: \(\phi(H a)=K a\). Prove the following : $$ \phi \text { is a homomorphism. } $$

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Let \(T\) designate the set \(\\{\) cis \(x: x \in \mathbb{R}\\}\), that is, the set of all the complex numbers lying on the unit circle, with the operation of multiplication. Use part 1 to prove that \(T\) is a group. ( \(T\) is called the circle group.)

If \(H\) is a subgroup of a group \(G\), let \(X\) designate the set of all the cosets of \(H\) in \(G\). For each element \(a \in G\), define \(\rho_{a}: X \rightarrow X\) as follows: $$ \rho_{a}(H x)=H(x a) $$ Prove that \(h: G \rightarrow S_{x}\) defined by \(h(a)=\rho_{a}\) is a homomorphism.

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : If \(H K=\\{x y: x \in H\) and \(y \in K\\}\), then \(H K\) is a subgroup of \(G\).

Let \(\alpha: \mathscr{F}(\mathbb{R}) \rightarrow \mathbb{R}\) be defined by \(\alpha(f)=f(1)\) and let \(\beta: \mathscr{F}(\mathbb{R}) \rightarrow \mathbb{R}\) be defined by \(\beta(f)=f(2)\) Let \(J\) be the set of all the functions from \(\mathbb{R}\) to \(\mathbb{R}\) whose graph passes through the point \((1,0)\) and let \(K\) be the set of all the functions whose graph passes through \((2,0)\). Use the \(\mathrm{FHT}\) to prove that \(\mathbb{R} \cong \mathscr{F}(\mathbb{R}) / J\) and \(\mathbb{R} \cong \mathscr{F}(\mathbb{R}) / K\).

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. $$ \mathbb{Z}_{3} \text { and } \mathbb{Z}_{6} /\langle 3\rangle $$

In the remaining exercises of this set, let \(p\) be a prime number, and assume \(G\) is a finite abelian group such that the order of every element in \(G\) is some power of \(p\). Let \(a \in G\) be an element whose order is the highest possible in \(G .\) We will argue by induction to prove that \(G\) is "decomposable." Let \(H=\langle a\rangle\). Explain why we may assume that \(G / H=\left[H b_{1}, \ldots, H b_{n}\right]\) for some \(b_{1}, \ldots, b_{n} \in G\).

Let \(G_{p^{t}}\) be the subgroup of \(G\) consisting of all elements whose order divides \(p^{k} .\) Let \(G_{m}\) be the subgroup of \(G\) consisting of all elements whose order divides \(m\). Prove: $$ G_{p^{k}} \cap G_{m}=\\{e\\} $$

Let \(f\) be a homomorphism from \(G\) onto \(H\) with kernel \(K\) : $$ G \underset{K}{R} H $$ If \(S\) is any subgroup of \(H\), let \(S^{*}=\\{x \in G: f(x) \in S\\}\). Prove: Let \(g\) be the restriction of \(f\) to \(S^{*}\). [That is, \(g(x)=f(x)\) for every \(x \in S^{*}\), and \(S^{*}\) is the domain of \(g\). ] Then \(g\) is a homomorphism from \(S^{*}\) onto \(S\), and \(K=\operatorname{ker} g\).

Let \(H\) and \(K\) be normal subgroups of a group \(G\), with \(H \subseteq K\). Define \(\phi: G / H \rightarrow G / K\) by: \(\phi(H a)=K a\). Prove the following : $$ \phi \text { is suriective. } $$

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Prove that \(f(x)=\) cis \(x\) is a homomorphism from \(\mathbb{R}\) onto \(T\).

If \(H\) is a subgroup of a group \(G\), let \(X\) designate the set of all the cosets of \(H\) in \(G\). For each element \(a \in G\), define \(\rho_{a}: X \rightarrow X\) as follows: $$ \rho_{a}(H x)=H(x a) $$ Prove that the set \(\left\\{a \in H: x a x^{-1} \in H\right.\) for every \(\left.x \in G\right\\}\), that is, the set of all the elements of \(H\) whose conjugates are all in \(H\), is the kernel of \(h\).

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : \(H\) is a normal subgroup of \(H K\).

Let \(G\) be an abelian group. Let \(H=\left\\{x^{2}: x \in G\right\\}\) and \(K=\left\\{x \in G: x^{2}=e\right\\}\). Use the FHT to conclude that \(H \cong G / K\).

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. $$ \mathbb{Z}_{2} \text { and } S_{3} /\\{\varepsilon, \beta, \delta\\} $$

Let \(f\) be a homomorphism from \(G\) onto \(H\) with kernel \(K\) : $$ G \underset{K}{R} H $$ If \(S\) is any subgroup of \(H\), let \(S^{*}=\\{x \in G: f(x) \in S\\}\). Prove: $$ S \cong S^{*} / K $$

Let \(H\) and \(K\) be normal subgroups of a group \(G\), with \(H \subseteq K\). Define \(\phi: G / H \rightarrow G / K\) by: \(\phi(H a)=K a\). Prove the following : $$ \text { ker } \phi=K / H $$

If \(H\) is a subgroup of a group \(G\), let \(X\) designate the set of all the cosets of \(H\) in \(G\). For each element \(a \in G\), define \(\rho_{a}: X \rightarrow X\) as follows: $$ \rho_{a}(H x)=H(x a) $$ Prove that if \(H\) contains no normal subgroup of \(G\) except \(\\{e\\}\), then \(G\) is isomorphic to a subgroup of \(S_{x}\).

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : Every member of the quotient group \(H K / H\) may be written in the form \(H k\) for some \(k \in K\).

Let \(G, M\), and \(N\) be groups, let \(f: G \rightarrow M\) be a homomorphism from \(G\) onto \(M\), and let \(h: G \rightarrow N\) be a homomorphism from \(G\) onto \(N\). Show that \(\phi(x)=(f(x), h(x))\) is a homomorphism from \(G\) onto \(M \times N\).

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. \(P_{2}\) and \(P_{3} / K\), where \(K=\\{\phi,\\{3\\}\\} .\) [HINT: Consider the function \(f(C)=\) \(C \cap\\{1,2\\} . P_{3}\) is the group of subsets of \(\\{1,2,3\\}\), and \(P_{2}\) of \(\left.\\{1,2\\} .\right]\)

Let \(H\) and \(K\) be normal subgroups of a group \(G\), with \(H \subseteq K\). Define \(\phi: G / H \rightarrow G / K\) by: \(\phi(H a)=K a\). Prove the following : $$ \text { Conclude (using the FHT) that }(G / H) /(K / H) \cong G / K \text {. } $$

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Use the FHT to conclude that \(T \cong \mathbb{R} /\langle 2 \pi\rangle\).

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : The function \(f(k)=H k\) is a homomorphism from \(K\) onto \(H K / H\), and its kernel is \(H \cap K\).

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. \(\mathbb{Z}_{3}\) and \(\left(\mathbb{Z}_{3} \times \mathbb{Z}_{3}\right) / K\), where \(K=\\{(0,0),(1,1),(2,2)\\}\). [HINT: Consider the function \(f(a, b)=a-b\) from \(\mathbb{Z}_{3} \times \mathbb{Z}_{3}\) to \(\mathbb{Z}_{3}\).]

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Prove that \(g(x)=\) cis \(\pi x\) is a homomorphism from \(\mathbb{R}\) onto \(T\), with kernel \(\mathbb{Z}\).

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : By the FHT, \(H /(H \cap K) \cong H K / H\). (This is referred to as the first isomorphism theorem.)

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Conclude that \(T \cong \mathbb{R} / \mathbb{Z}\).