Chapter 14

If \(G\) is any group, let \(H\) and \(K\) be normal subgroups of \(G\) such that \(H \cap K=\\{e\\}\). Prove the following: \(\mathbf{1}\) Let \(h_{1}\) and \(h_{2}\) be any two elements of \(H\), and \(k_{1}\) and \(k_{2}\) any two elements of \(K\). $$ h_{1} k_{1}=h_{2} k_{2} \quad \text { implies } \quad h_{1}=h_{2} \quad \text { and } \quad k_{1}=k_{2} $$ (HINT: If \(h_{1} k_{1}=h_{2} k_{2}\), then \(h_{2}^{-1} h_{1} \in H \cap K\) and \(k_{2} k_{1}^{-1} \in H \cap K\). Explain why.)

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If \(G\) is abelian, then \(H\) is abelian.

If \(f: G \rightarrow H\) is a homomorphism, prove each of the following: For each element \(a \in G\), the order of \(f(a)\) is a divisor of the order of \(a\).

In the following, let \(G\) denote an arbitrary group. Find all the normal subgroups (a) of \(S_{3}\) (b) of \(D_{4}\). Prove the following:

Let \(G, H\), and \(K\) be groups. Prove the following: If \(f: G \rightarrow H\) and \(g: H \rightarrow K\) are homomorphisms, then their composite \(g \circ f\) : \(G \rightarrow K\) is a homomorphism.

Prove that each of the following is a homomorphism, and describe its kernel. The function \(\phi: \mathscr{F}(\mathbb{R}) \rightarrow \mathbb{R}\) given by \(\phi(f)=f(0)\).

If \(G\) is any group, let \(H\) and \(K\) be normal subgroups of \(G\) such that \(H \cap K=\\{e\\}\). Prove the following: For any \(h \in H\) and \(k \in K, h k=k h .\) (HINT: \(h k=k h\) iff \(h k h^{-1} k^{-1}=e .\) Use the fact that \(H\) and \(K\) are normal.)

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If \(G\) is cyclic, then \(H\) is cyclic.

If \(f: G \rightarrow H\) is a homomorphism, prove each of the following: The order of any element \(b \neq e\) in the range of \(f\) is a common divisor of \(|G|\) and | H|. [Use (1).]

Let \(G\) denote a group, and \(H\) a subgroup of \(G\). Prove the following: Suppose an element \(a \in G\) has order \(2 .\) Then \(\langle a\rangle\) is a normal subgroup of \(G\) iff \(a\) is in the center of \(G\).

In the following, let \(G\) denote an arbitrary group. Every subgroup of an abelian group is normal.

Let \(G, H\), and \(K\) be groups. Prove the following: If \(f: G \rightarrow H\) is a homomorphism with kernel \(K\), then \(f\) is injective iff \(K=\\{e\\}\).

Let \(H\) be a subgroup of \(G\). For any \(a \in G\), let \(a H a^{-1}=\left\\{a \times a^{-1}: x \in H\right\\} ; a H a^{-1}\) is called a conjugate of \(H .\) Prove the following: \(H\) is a normal subgroup of \(G\) iff \(H=a H a^{-1}\) for every \(a \in G\). In the remaining exercises of this set, let \(G\) be a finite group. By the normalizer of \(H_{1}\) we mean the set \(N(H)=\left\\{a \in G: a x a^{-1} \in H\right.\) for every \(\left.x \in H\right\\}\).

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If every element of \(G\) has finite order, then every element of \(H\) has finite order.

If \(f: G \rightarrow H\) is a homomorphism, prove each of the following: If the range of \(f\) has \(n\) elements, then \(x^{n} \in \operatorname{ker} f\) for every \(x \in G\).

Let \(G\) denote a group, and \(H\) a subgroup of \(G\). Prove the following: If \(a\) is any element of \(G,\langle a\rangle\) is a normal subgroup of \(G\) iff \(a\) has the following property: For any \(x \in G\), there is a positive integer \(k\) such that \(x a=a^{k} x\).

In the following, let \(G\) denote an arbitrary group. The center of any group \(G\) is a normal subgroup of \(G\).

Let \(G, H\), and \(K\) be groups. Prove the following: If \(f: G \rightarrow H\) is a homomorphism and \(K\) is any subgroup of \(G\), then \(f(K)=\\{f(x): x \in K\\}\) is a subgroup of \(H\)

Find a homomorphism \(f: \mathbb{Z}_{15} \rightarrow \mathbb{Z}_{5}\), and indicate its kernel. (Do not actually verify that \(f\) is a homomorphism.)

Let \(H\) be a subgroup of \(G\). For any \(a \in G\), let \(a H a^{-1}=\left\\{a \times a^{-1}: x \in H\right\\} ; a H a^{-1}\) is called a conjugate of \(H .\) Prove the following: If \(a \in N(H)\), then \(a H a^{-1}=H\). (Remember that \(G\) is now a finite group.)

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If every element of \(G\) is its own inverse. every element of \(H\) is its own inverse

If \(f: G \rightarrow H\) is a homomorphism, prove each of the following: Let \(m\) be an integer such that \(m\) and \(|H|\) are relatively prime. For any \(x \in G\), if \(x^{m} \in \operatorname{ker} f\), then \(x \in\) ker \(f\)

Let \(G\) denote a group, and \(H\) a subgroup of \(G\). Prove the following: In a group \(G\), a commutator is any product of the form \(a b a^{-1} b^{-1}\), where \(a\) and \(b\) are any elements of \(G .\) If a subgroup \(H\) of \(G\) contains all the commutators of \(G\), then \(H\) is normal.

Let \(G, H\), and \(K\) be groups. Prove the following: If \(f: G \rightarrow H\) is a homomorphism and \(J\) is any subgroup of \(H\), then- $$ f^{-1}(J)=\\{x \in G: f(x) \in J\\} $$ is a subgroup of \(G\). Furthermore, ker \(f \subseteq f^{-1}(J)\).

Imagine a square as a piece of paper lying on a table. The side facing you is side A. The side hidden from view is side \(B\). Every motion of the square either inter- changes the two sides (that is, side \(B\) becomes visible and side \(A\) hidden) or leaves the sides as they were. In other words, every motion \(R_{i}\) of the square brings about one of the permutations $$ \left(\begin{array}{ll} A & B \\ A & B \end{array}\right) \quad \text { or } \quad\left(\begin{array}{ll} A & B \\ B & A \end{array}\right) $$ of the sides; call it \(g\left(R_{i}\right)\). Verify that \(g: D_{4} \rightarrow S_{2}\) is a homomorphism, and give its kernel.

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If every element of \(G\) has a square root, then every element of \(H\) has a square root.

Let \(G\) denote a group, and \(H\) a subgroup of \(G\). Prove the following: If \(H\) and \(K\) are subgroups of \(G\), and \(K\) is normal, then \(H K\) is a subgroup of \(G\). (HK denotes the set of all products \(h k\) as \(h\) ranges over \(H\) and \(k\) ranges over \(K .\) )

In the following, let \(G\) denote an arbitrary group. Let \(H\) be a subgroup of \(G . H\) is normal iff \(a H=H a\) for every \(a \in G\).

Let \(G, H\), and \(K\) be groups. Prove the following: If \(f: G \rightarrow H\) is a homomorphism with kernel \(K\), and \(J\) is a subgroup of \(G\), let \(f_{J}\) designate the restriction of \(f\) to \(J\). (In other words, \(f_{J}\) is the same function as \(f\), except that its domain is restricted to \(J\).) Prove that ker \(f_{J}=J \cap K\).

Every motion of the regular hexagon brings about a permutation of its diagonals, labeled 1,2, and \(3 .\) For each \(R_{i} \in D_{6}\), let \(f\left(R_{i}\right)\) be the permutation of the diagonals produced by \(R_{i} .\) Argue informally (appealing to geometric intuition) to explain why \(f: D_{6} \rightarrow S_{3}\) is a homomorphism. Then complete the following: $$ f\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 & 4 & 5 & 6 \end{array}\right)=\varepsilon \quad f\left(\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 5 & 6 & 1 \end{array}\right)=\delta \quad \ldots $$ (That is, find the value of \(f\) on all 12 elements of \(D_{6} .\) )

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If \(G\) is finitely generated, then \(H\) is finitely generated. (A group is said to be "finitely generated" if it is generated by finitely many of its elements.)

Let \(G\) denote a group, and \(H\) a subgroup of \(G\). Prove the following: Let \(S\) be the union of all the cosets \(\mathrm{Ha}\) such that \(H a=a H\). Then \(S\) is a normal subgroup of \(G\).

In the following, let \(G\) denote an arbitrary group. Any intersection of normal subgroups of \(G\) is a normal subgroup of \(G\).

Let \(G, H\), and \(K\) be groups. Prove the following: For any group \(G\), the function \(f: G \rightarrow G\) defined by \(f(x)=e\) is a homomorphism.

Prove that each of the following is a homomorphism, and describe its kernel. Let \(G\) be the multiplicative group of all \(2 \times 2\) matrices $$ \left(\begin{array}{ll} a & b \\ c & d \end{array}\right) $$ satisfying \(a d-b c \neq 0\). Let \(f: G \rightarrow \mathbb{R}^{*}\) be given by \(f(A)=\) determinant of \(A=a d-b c\)

Let \(G, H\), and \(K\) be groups. Prove the following: For any group \(G,\\{e\\}\) and \(G\) are homomorphic images of \(G\).

Let \(G, H\), and \(K\) be groups. Prove the following: The function \(f: G \rightarrow G\) defined by \(f(x)=x^{2}\) is a homomorphism iff \(G\) is abelian.

Let \(G, H\), and \(K\) be groups. Prove the following: The functions \(f_{1}(x, y)=x\) and \(f_{2}(x, y)=y\), from \(G \times H\) to \(G\) and \(H\), respectively, are homomorphisms.