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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