Math 360, Fall 2021, Assignment 6

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

Read:

1. Section 5.

Carefully define the following terms, then give one example and one non-example of each:

1. Symmetry (of a subset $A\subseteq\mathbb{R}^n$).
2. Symmetry group (of a subset $A\subseteq\mathbb{R}^n$).
3. Order (of a group; see Definition 5.3 on page 50 of the text).
4. $D_n$ (the dihedral group with order $2n$).
5. Subgroup (of a group).
6. Trivial subgroup (see Definition 5.5 on page 51 of the text).
7. Improper subgroup (see Definition 5.5 on page 51 of the text).
8. $\left\langle S\right\rangle$ (the subgroup generated by the subset $S$).
9. $\left\langle g\right\rangle$ (the cyclic subgroup generated by the element $g$; see Definition 5.18 on page 54).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning unions and intersections of subgroups.

Solve the following problems:

1. Section 5, problems 1, 2, 8, 9, 11, 12, 21, 22, 23, 24, 25, and 36.
2. Prove that $(\mathbb{R},+)$ is not a cyclic group. (Hint: $\mathbb{R}$ is an uncountable set. Now look again at the list of elements of a cyclic subgroup. What can you conclude about the cardinality of a cyclic group?)

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Definitions:

1. Symmetry: the property that a mathematical object remains unchanged under a set of operations or transformations. Suppose $A \subseteq R^{n}$. A symmetry of $A$ is an isometry $\sigma : R^{n} \rightarrow R^{n}$ such that if $x \in A$ then also $\sigma (x) = A$. Claims: (1) The composition of two symmetries of $A$ is again a symmetry of $A$. (2) The identity map is always a symmetry of $A$ ("trivial symmetry"). (3) Any symmetry is invertible, and its inverse is again a symmetry.
2. Symmetry group: Suppose $A\subseteq\mathbb{R}^n$. The symmetry group of $A$ is the set of symmetries of $A$ (see above), regarded as a group under composition. (Not to be confused with the symmetric group $\mathrm{Sym}(A)$, which is the group of all bijections from $A$ to $A$ and is usually a far larger group than the symmetry group defined here.)
3. If $G$ is a group, then the order $|G|$ of $G$ is the number of elements in $G$. (Recall from Section 0 that, for any set S, $|S|$ is the cardinality of $S$.)
4. The symmetry group of a regular n-gon.
5. Subgroup: Suppose $(G, \triangle )$ is a group, and $H \subseteq G$. We say that $H$ is a subgroup of $G$ if (1) $e \in H$, (2) if $h_{1}, h_{2} \in H$, then $h_{1} \triangle h_{2} \in H$, (3) if $h \in H$, then $h' \in H$. (This ensure that $(H, \triangle )$ is also a group).
6. If $G$ is a group, The subgroup $\{e\}$ is the trivial subgroup of G. All other subgroups are nontrivial.
7. If $G$ is a group, then the subgroup consisting of $G$ itself is the improper subgroup of $G$. All other subgroups are proper subgroups.
8. Suppose $(G, \triangle )$ is a group, and $S \subseteq G$ (subset, not necessarily a subgroup). define $\left\langle S \right\rangle$ to be the intersection of all subgroups of $G$ that contain $S$.
9. let $G$ be a group and let $g \in G$. then the subgroup $\{ g^n | n\in \mathbb Z \}$ of $G$, characterized in Theorem 5.17, is called the cyclic subgroup of $G$ generated by $g$, and denote by $\left\langle g \right\rangle$. Theorem 5.17: $H = \{ g^n | n \in \mathbb Z \}$ is a subgroup of $G$ and is the smallest subgroup go $G$ that contain $g$, that is, every subgroup containing $g$ contains $H$.

Theorems:

1. Suppose $(G, \triangle )$ is a group, and $H \leq G$ (is subgroup of) and $K \leq G$. Then $H \cap K$ is also a subgroup (however, $H \cup K$ is usually not).

Book Problems:

1. 1. For $\mathbb R \in \mathbb C$, under addition, (1) $e = 0 \in \mathbb C$, $0 \in \mathbb R, e \in \mathbb R$. (2) If $a, b \in \mathbb R,a + b \in \mathbb R$. (3) If $a \in \mathbb R$, $a' = -a \mathbb R$. Yes, it is a group.
2. 2. No. (1) $0 \not\in \mathbb Q^{+}$, (3)For $a \in \mathbb Q^{+}, a' = -a < 0 \not\in \mathbb Q^{+}$.
3. 8. No. Suppose this is a set $A$. (1)$\det(e) = 1 \neq 2, e \not\in A$. Inverse not in it. (2)$det(M) * det(N) = 4 \neq 2, \forall M, N \in A$, not closed under matrix multiplication.
4. 9. Yes. $I=\mathrm{diag}(1,1,\dots,1)$ is a diagonal matrix without zeros on the diagonal. If $A=\mathrm{diag}(a_1,\dots,a_n)$ and $B=\mathrm{diag}(b_1,\dots,b_n)$ are two such matrices, then so it $AB=\mathrm{diag}(a_1b_1,\dots,a_nb_n)$, as is $A^{-1}=\mathrm{diag}(a_1^{-1},\dots,a_n^{-1})$. (Note that although $\{(1, 1), (1, 2) \}$ has determinant $0$, is it not a diagonal matrix.)
5. 11. No: the identity matrix does not belong to this set.
6. 12. Yes: the identity matrix belongs to this set, and if $\det(A)=\pm1$ and $\det(B)=\pm1$ then also $\det(AB)=\pm1$ and $\det(A^{-1})=\pm1$.
7. 21. (a) $\{ -50, -25, 0, 25, 50 \}$. (b) $\{4, 2, 1, \frac{1}{2}, \frac{1}{4} \}$. (c) $\{ \frac{1}{\pi ^2}, \frac{1}{\pi}, 1, \pi, \pi^2 \}$.
8. 22. $\left\{\begin{bmatrix}0&-1\\-1&0\end{bmatrix},\begin{bmatrix}1&0\\0&1\end{bmatrix}\right\}$.
9. 23. $\left\{\left.\begin{bmatrix}1&n\\0&1\end{bmatrix}\,\right\rvert\,n\in\mathbb{Z}\right\}$.
10. 24. $\left\{\left.\begin{bmatrix}3^n&0\\0&2^n\end{bmatrix}\,\right\rvert\,n\in\mathbb{Z}\right\}$.
11. 25. $\left\{\dots,\begin{bmatrix}0&-1/2\\-1/2&0\end{bmatrix},\begin{bmatrix}1&0\\0&1\end{bmatrix},\begin{bmatrix}0&-2\\-2&0\end{bmatrix},\begin{bmatrix}4&0\\0&4\end{bmatrix},\begin{bmatrix}0&-8\\-8&0\end{bmatrix},\dots\right\}$.
12. 36. (a) see http://jwilson.coe.uga.edu/EMAT6680/Parsons/MVP6690/Essay1/modular.html (b) $\{0\}, \mathbb Z_6, \{0, 2, 4\}, \{0, 3\}, \{0, 2, 4\}, \mathbb Z_6$. (c)$1, 5$ (d)See https://drive.google.com/file/d/1ZU7k9peVMdC_mY5XCtIT5-Ld8Z4k_bjy/view?usp=sharing

Other problems:

By the classification of cyclic groups, and cyclic group must be isomorphic with either $(\mathbb{Z},+)$ or with $(\mathbb{Z}_n,+_n)$. In particular, any cyclic group must be equinumerous either with $\mathbb{Z}$ or with $\mathbb{Z}_n$. But $\mathbb{R}$ is an uncountable set, so $(\mathbb{R},+)$ cannot be cyclic.