Math 360, Fall 2019, Assignment 6

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

Read:

1. Section 5.

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

1. Subgroup (of a group).
2. $\left\langle S\right\rangle$ (the subgroup generated by the subset $S$).
3. Cyclic group.

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

1. Theorem concerning the number of solutions of the equation $a*x=b$.
2. Theorem concerning the number of solutions of the equation $x*a=b$.
3. Theorem concerning the number of occurrences of an element in a row or column of a group table.
4. Theorem concerning groups with two elements.
5. Theorem concerning groups with three elements.
6. Theorem concerning unions and intersections of subgroups.
7. Theorem characterizing when $\left\langle S\right\rangle\subseteq H$ (when $H$ is known to be a subgroup).

Solve the following problems:

1. Section 5, problems 1, 2, 9, 11, 21, 22, 23, and 27.
2. Prove that every cyclic group is abelian. (Hint: suppose $G$ is cyclic, say $G=\left\langle a\right\rangle$. Then every element of $G$ can be written in the form $a^i$ for some $i\in\mathbb{Z}$. Now use the laws of exponents.)
3. In class we gave a rather tedious direct verification that the symmetry group of an equilateral triangle is not cyclic. Now demonstrate that fact with a one-line argument.
4. Prove that $GL_n(\mathbb{R})$ is never cyclic, assuming that $n>1$. (Next week we will show that $GL_1(\mathbb{R})$ is not cyclic either.)

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