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 $\PageIndex{8}$

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 $\PageIndex{9}$

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 $\PageIndex{10}$

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 $\PageIndex{11}$

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

Exercise $\PageIndex{12}$

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

Exercise $\PageIndex{13}$

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

Exercise $\PageIndex{14}$

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

Exercise $\PageIndex{15}$

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 $\PageIndex{16}$

Prove or disprove: If $K$ is a subgroup of $H$ and $H$ is a subgroup of $G$ then $K$ is a subgroup of $G$.