2E: Exercises

Exercise \(\PageIndex{1}\)

Check the following structures for the group properties:

1. \(\mathbb{Z}_4 \) under addition.

2. \(\mathbb{Z}_5^* \) under multiplication.

3. \( \mathbb{Z}_8^* \) under multiplication.

4. \(\{ \pm 1, \pm i\} \) under multiplication.

5. \(\mathbb{R}, \star\) where \(a \star b=a+b+3\) for \(a,b \in \mathbb{R}\).

6. The four functions \(f(x)=x, g(x)=-x, h(x)= \dfrac{1}{x}, j(x)= \dfrac{-1}{x}\) under composition.

7. The set of all even integers under addition.

8. The four matrices \( \begin{bmatrix}
   1 & 0 \\
   0 & 1
   \end{bmatrix}, 
   \begin{bmatrix}
   -1 & 0 \\
   0 & -1
   \end{bmatrix},
   \begin{bmatrix}
   1 & 0 \\
   0 & -1
   \end{bmatrix},
   \begin{bmatrix}
   -1 & 0 \\
   0 & 1
   \end{bmatrix}
   \) under multiplication.

Exercise \(\PageIndex{2}\)

Let \(H(C) =\left\{ \begin{bmatrix}1 & a & b\\
0 & 1 & c \\
0 & 0 & 1 \end{bmatrix}\Big| \ a,b,c, \in \mathbb{C}\right\}\).

Show that \(H(\mathbb{C})\) is a group under matrix multiplication.  Demonstrate explicitly that \(H(\mathbb{C})\) is always non-abelian.

Exercise \(\PageIndex{3}\)

Let \(S = \mathbb{R} \backslash \{-1\}\) and define a binary operation \(\oplus\) on \(S\) by \(a \oplus b = a+b+ab\).  Prove that \((S, \oplus)\) is an abelian group.

Exercise \(\PageIndex{4}\)

Given the groups \(\mathbb{R}^*\) and \(\mathbb{Z}\), let \(G = \mathbb{R}^* \times \mathbb{Z}\).  Define a binary operation \(\star\) by \((a,m) \star (b, n) = (ab,m + n)\). Show that \( (G, \star)\) is a group under this operation.

Exercise \(\PageIndex{5}\)

Let \(G\) be a group. 

1. Show that  Let \(a,b , c \in G\). If \(ab=ac\) then \(b=c\). Why is the analogous cancellation rule false for arbitrary matrices?

2. Show that if \(a^2 = e\), for all elements of \(a \in G\), then \(G\) must be abelian. Give an example to show that the converse of the above statement is not true.

3. Let \(a,b  \in G\).  Solve \(axb=b\) for \(x\); show the order in which you multiply. 
4.  Let \(a,b  \in G\).  Prove that \((ab)^{-1}=b^{-1}a^{-1}\), without assuming that \(G\) is abelian. Explain why the proposed identity \((ab)^{-1}=a^{-1}b^{-1}\)is not valid for every group. A specific counterexample or an explanation using noncommutativity is acceptable.

 

Exercise \(\PageIndex{6}\)

Let \(H\) consists of \(2 \times 2\) matrices of the form \(\begin{bmatrix} \cos{(x)} & -\sin{(x)}\\
\sin{(x)} & \cos{(x)} \end{bmatrix}\), where \(x \in \mathbb{R}\). Prove that \(H\) is a subgroup of \(SL_2(\mathbb{R})\).

Exercise \(\PageIndex{7}\)

Prove or disprove the following statements: Let \(H\) and \(K\) be subgroups of a group \(G\).

1. \(H \cup K\) is a subgroup of \(G\).

2. \(H \cap K\) is a subgroup of \(G\).

3. Prove that \(gHg^{-1} = \{ ghg^{-1} \mid h \in H \}\) is a subgroup of \(G\).

 

Exercise \(\PageIndex{8}\)

Consider the group \(\mathbb{Z}_{14}\) under addition.

1. Find the order of every element.

2. Find all the subgroups.

Exercise \(\PageIndex{9}\)

Prove that for each element \(a \in G\), where \(G\) is a group, the centralizer of \(a\), \(C(a)\) is a subgroup of \(G\). Prove that for each element \(a \in G\), where \(G\) is a group, that \(C(a)=C(a^{-1})\).

Exercise \(\PageIndex{10}\)

Let \(G=GL(2,\mathbb{R})\).  Then find

1. \(C \Bigg(\begin{bmatrix}1 & 1\\
   1 & 0 \end{bmatrix}\Bigg)\).

2. \(C \Bigg( \begin{bmatrix} 0 & 1\\
   1 & 0 \end{bmatrix}\Bigg)\).

3. \(\mathbb{Z}(G)\).

Exercise \(\PageIndex{11}\)

Let \(G\) be a group. \(H=\{g^2: g\in G\}. \)

1. Prove or disprove: If \(G\) is abelian, then  \(H\) is a subgroup.

2. Prove or disprove: If \(H\) is a subgroup, then \(G\) is abelian.

Exercise \(\PageIndex{12}\)

List the cyclic subgroups of \(U(30)=\{1,7,11,13,17,19,23,29\} \).

Exercise \(\PageIndex{13}\)

Show that \(U(20)\ne <k> \) for any \(k \) in \(U(20) \).  [Hence, \(U(20) \) is not cyclic.]

Exercise \(\PageIndex{14}\)

Decide whether \(U(10) \) is cyclic or not. Justify your answer.

Exercise \(\PageIndex{15}\)

Is \((\mathbb{Z},+) \) cyclic group?  If so, what are the possible generators?

Exercise \(\PageIndex{16}\)

Decide whether the following \(H\) is a subgroup of \(G \). Justify your answer. If it is a subgroup, is it abelian?

1. \(H=\mathbb{N}, G=\mathbb{Z}\)
2. \(H=\{1,3\}, G=\mathbb{Z}^*_{10}\)
3. \(H=\{1,3\}, G=\mathbb{Z}^*_{15}\)
4. \(H=\{ \left. (n,m) \in \mathbb{Z}\times \mathbb{Z}  \right| \,\, 2\mid (n+m) \}, G=\mathbb{Z}\times \mathbb{Z}\)
5. \(H = \left\{ \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \;\middle|\; a \in \mathbb{R} \right\},  G=SL_2(\mathbb{R})\).
6. \(H = \left\{ A \in GL_2(\mathbb{R}) \;\middle| \det(A)>0 \right\},  G=GL_2(\mathbb{R})\).

Exercise \(\PageIndex{17}\)

Prove or disprove: If \(K\) is a subgroup of \(H\) and \(H\) is a subgroup of \(G\) then \(K\) is a subgroup of \(G\).

Exercise \(\PageIndex{18}\)

Let \(G = \langle a \rangle\) be a cyclic group generated by an element \(a \in G\). Suppose \(G\) is infinite cyclic. Prove that \(a\) and \(a^{-1}\) are the only generators of \(G\).

 

Exercise \(\PageIndex{19}\)

In \(\mathbb{Z}_8\) under addition, solve \(5+x=2\) and find the inverse of \(5\).