Concerning Cyclic Groups

Question

I am new to group theory. I have a problem but I don't really understand what it is about, so I am asking somebody to explain what is the problem (I am not really seeking for solution).

Here it is:

Let $G = C_{20}$ be THE cyclic group of order $20$. Find the number of (distinct) solutions of the equation $x^n=1$ in the group $G$, when $n=6,7,8,9$ and $10$.

First of all, is there only one cyclic group of order $20$?

It doesn't say what are the elements of the group (are they integers?), what is the operation of the group. Is there anything missing from the problem description?

Answer 1

The question is well posed. The reason the question refers to the cyclic group of order $20$ is that all such groups are isomorphic. They are identical in terms of their group-theoretic properties.

Therefore the answer to the question will be the same whether you answer it for one cyclic group of order $20$ or any other. In this case, you can select any cyclic group of order $20$, whichever is most convenient.

In most circumstances, it will be most convenient to select the group $\mathbf{Z}/20\mathbf{Z}$. Its elements are the equivalences classes of the equivalence relation on $\mathbf{Z}$ of equivalence modulo $20$. This is probably what your question denotes by $C_{20}$.

Answer 2

There are multiple ways of representing a group. Typically, if two groups have the same properties (isomorphic) they are thought of as the same, and you can pass between the different ways of viewing the problem freely. I would say that $\mathbb Z /5$ has 3 as an element because you can think of it as 0,1,2,3,4 where values greater than 5 are removed. Or you can think of it as congruence classes. Sometimes the different ways of thinking abour a group are not as natural, but it is not unreasonable to consider the groups to still be the same. Since they have the same group properties, they are the same. It is true that there is only one cyclic group of a given order. It can be represented in different ways. So one way you can think about it is if you had integers and "moded," out. But the elements of Z/n are really the congruence classes of integers (i.e. 2,7,12 ... are in the congruence class $\overline 2$ with respect to 5). But these congruences classes it turns out act in pretty much the exact same way as integers for this purpose.

Answer 3

The order of the elements of a group is a divisor of the order of the order of the group. So using this theorem you can find the number of distinct solutions of the equation $x^n=1$ for given n.

Answer 4

Nice question, as it was clarified above you may choose G in question to be isomorphic to $\mathbb{Z}/20\mathbb{Z}$.

Now by Langrange's Theorem you have that the order of any element $\alpha \in G$ divides the order of G. This follows when you notice that $\langle \alpha \rangle$ is a subgroup of G. In other words $$\alpha^{|G|}=e$$

This might guide you finding the solution for $\alpha^{n}=\overline1$.