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. Rotations that map a brick to itself.

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

7. 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.

Answer

5. If \(a,b,c\) denote rotations about three mutually perpendicular axes, then \(a^2=b^2=c^2=e\) the identity rotation. Also \(ab=c,ac=b,bc=a\) Thus this group is isomorphic to \(K_4.\)

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.  Show that if \(a^2 = e\), for all elements of \(a \in G\), then \(G\) must be abelian.

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 \(G\).

 

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.

1. \(H=\mathbb{N}, G=\mathbb{Z}\)
2. \(H=\{1,3\}, G=\mathbb{Z}^*_{10}\)
3. \(H=\{1,3\}, G=\mathbb{Z}^*_{15}\)
4. \(H=\{ \left. (n,m) \in \mathbb{Z}\times \mathbb{Z}  \right| \,\, 2\mid (n+m) \}, G=\mathbb{Z}\times \mathbb{Z}\)

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 \(G\).