# 2E: Exercises

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

1. 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.

1. 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.

2. 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.

3. Let $$G$$ be a group.  Show that if $$a^2 = e$$, for all elements of $$a \in G$$, then $$G$$ must be abelian.

1. 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 $$SL_2(\mathbb{R})$$.

1. 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 $$G$$.

2. 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 $$C(a)=C(a^{-1})$$.

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

1. 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.

1. List the cyclic subgroups of $$U(30)$$.

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

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

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

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

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

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

2. 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.