# 2E: Exercises

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##### 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.

##### Exercise $$\PageIndex{2}$$

Let

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

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 $$\PageIndex{11}$$
1. List the cyclic subgroups of $$U(30)$$.

$$U(30)=\{1,7,11,13,17,19,23,29\}$$.

Thus $$|U(30)|=8$$.

##### Exercise $$\PageIndex{12}$$

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

$$U(20)=\{1,3,7,9,11,13,17,19\}$$.

$$|U(20)|=8$$.

##### Exercise $$\PageIndex{13}$$

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

##### Exercise $$\PageIndex{14}$$

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

This page titled 2E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.