Math 360, Fall 2021, Assignment 7

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

Read:

1. Section 6.

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

1. Multiplicative notation (for a general group).
2. Additive notation (for a general abelian group).
3. Cyclic group.
4. Generator (of a cyclic group).

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

1. Laws of exponents.
2. Laws of multiples (i.e. the restatement of the laws of exponents in additive notation).
3. Theorem concerning integer division.
4. Classification of cyclic groups.

Solve the following problems:

1. Section 6, problems 1, 3, 9, 10, 17, 19, 33, 34, 35, 36, and 37.
2. Prove that every cyclic group is abelian. (Hint: every element has the form $g^i$ for some fixed generator $g$; now use the laws of exponents.)
3. Prove that every cyclic group is countable (i.e. either finite or countably infinite; you may utilize the classification of cyclic groups even though we have not yet completed its proof in class).
4. Show that each of the following subgroups of $(\mathbb{Z},+)$ can be generated by a single non-negative integer: (a) $\left\langle 4, 6\right\rangle$, (b) $\left\langle 15, 35\right\rangle$, and (c) $\left\langle 12, 18, 27\right\rangle$.
5. (Challenge) Following the pattern of the three parts of the last problem, try to guess a general formula for a single non-negative generator for the subgroup $\left\langle k_1,k_2,\dots,k_m\right\rangle$ of $(\mathbb{Z},+)$.

--------------------End of assignment--------------------

Definitions:

1. Multiplicative notation is a notational system in which the operation symbol is suppressed and the operation is indicated by juxtaposition. Thus, for example, in the group $(G,\triangle)$, the quantity $a\triangle b$ is abbreviated to $ab$. In this system the identity element is denoted by the symbol $1$. Also, $a^n$ means $aa\dots a$ ($n$ factors) when $n>0$, it means $a'a'\dots a'$ ($\left\lvert n\right\rvert$ factors) when $n<0$, and it means $1$ when $n=0$.
2. Additive notation is a notational system, reserved by convention for abelian groups, in which the operation is indicated by the symbol $+$. In this system the identity element is denoted $0$, and the quantity which would be written as $a^n$ in multiplicative notation is instead denoted $na$. # Suppose $G$ is a group. $G$ is said to be cyclic if there is some $g \in G$ with $\left\langle \{g\} \right\rangle = G$.
3. Let $G$ be a cyclic group. A generator for $G$ is an element $g\in G$ with $\left\langle g\right\rangle=G$.

Theorems:

1. Laws of Exponents: Suppose $G$ is any group, written multiplicatively. (1) $a^{i}a^{j} = a^{i+j}$, (2) $(a^{i})^{j} = a^{ij}$
2. $(ia) + (ja) = (i+j)a, j(ia) = (ji)a$
3. Suppose $a,b \in \mathbb Z$ and $b>0$. Then there exist unique $q, r \in \mathbb Z$, satisfying: $(1)a=bq+r, (2) 0 \leq r < b$.
4. Every cyclic group is isomorphic to $(\mathbb Z, +)$ or to $(\mathbb Z_n, +_n)$ for some n.

Textbook Solution:

1. 42 = 9·4+6, q = 4, r = 6

3. −50 = 8(−7)+6, q = −7, r = 6

Using the structure theory of finite cyclic groups, developed in subsequent assignments, it is not hard to prove that $[a]$ generates $\mathbb{Z}_n$ if and only if $\mathrm{gcd}(a,n)=1$, and this fact provides a convenient check for problems 9 and 10 below. However, it is not actually needed to answer these questions: it is straightforward to simply calculate all of the cyclic subgroups of the given groups and see which ones are improper. Similarly, the structure theory of finite cyclic groups provides convenient shortcuts for problem 17 but is not actually needed for it.

9. 1, 3, 5, and 7 are relatively prime to 8 so the answer is 4.

10. 1, 5, 7, and 11 are relatively prime to 12 so the answer is 4.

17. gcd(25, 30) = 5 and 30/5 = 6 so <25> has 6 elements, according to $g^{n}$.

19. The polar angle for i is π/2, so it generates a subgroup of 4 elements. $(e^{I\frac{\pi}{2}} = \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}) = i)$

33. The Klein 4-group, D4

34. $(\mathbb R, +)$

35. $\mathbb{Z}_2$

36. No such example exists. Every infinite cyclic group is isomorphic to $(\mathbb{Z}, +)$ which has just two generators, 1 and -1.

37. $\mathbb{Z}_8$ has generators 1, 3, 5, and 7.

Solution:

2. Let G be a cyclic group with a generator g∈G. Namely, we have G = ⟨g⟩ (every element in G is some power of g.). Let a and b be arbitrary elements in G. Then there exists n,m∈Z such that $a = g^n$ and $b = g^m$. It follows that

$ab = g^n g^m = g^{n+m} = g^{m+n} = g^m g^n = ba$

Hence we obtain ab = ba for arbitrary a,b∈G. Thus G is an abelian group.

3. Any cyclic group is isomorphic (hence also equinumerous) to either $\mathbb{Z}$ or to $\mathbb{Z}_n$, and both of these are countable.

4. For (a), note that $\left\langle 4,6\right\rangle=\{4x+6y\,|\,x,y\in\mathbb{Z}\}$, and any element of this set is evidently divisible by $2$, showing that $\left\langle4,6\right\rangle\subseteq\left\langle2\right\rangle$. On the other hand, $2=6-4$ is a member of $\left\langle4,6\right\rangle$, so we also have $\left\langle2\right\rangle\subseteq\left\langle4,6\right\rangle$ and thus in fact $\left\langle4,6\right\rangle=\left\langle2\right\rangle$. Similar arguments show that $\left\langle15,35\right\rangle=\left\langle5\right\rangle$ and $\left\langle12,18,27\right\rangle=\left\langle3\right\rangle$.

5. It would seem that $\left\langle k_1,\dots,k_n\right\rangle$ might be generated by the greatest common divisor of $k_1,\dots,k_n$.