How do I prove that $S=${$0,10,20$} is a subgroup of $\mathbb{Z_{30}}$?

Question

Prove that $S=${$0,10,20$} is a subgroup of $\mathbb{Z_{30}}$.

I would also like to know how to show that $\mathbb{Z_{30}}$ it's a group.

I know of theory that $\mathbb{Z_{n}}$ it's an abelian group but I do not understand why it is.

I'm having a very difficult time with such problems and would be very thankful to anyone who would take the time to slowly explain how does one even begin to take a stab at answering such a question. Thank you for your help.

Answer 1

Prove that S is a subgroup of $\Bbb Z_{30}$.

To do this - we apply the subgroup test:

• Closure. Take two elements from $S$. Then, is it true that the addition of these two elements are in $S$? Well, yes: $$0+10=10,\quad 0+20=20,\quad 10+20=0,$$ $$0+0=0,\quad 10+10=20,\quad 20+10=0.$$
• Identity. Is the identity element of $\Bbb Z_{30}$ present in $S$? Yes, $0$ is in $S$.
• Inverse. Given any element of $S$ - is its inverse in it also? Well, yes: $$0^{-1}=0,\quad 10^{-1}=20,\quad 20^{-1}=10.$$

So, $S$ is a subgroup of $\Bbb Z_{30}$.

I would also like to know how to show that $\Bbb Z_{30}$ is a group.

To do this - we must check the group axioms are satisfied for $\Bbb Z_{30}$:

• We find that closure and assosciativity will be apparent since these operations are closed when we add integers in general.
• The identity element in $\Bbb Z_{30}$ is $0$.
• For inverses, given any element in $\Bbb Z_{30}$, you should be able to write the corresponding number that, upon addition, gives the identity element $0$. For example, the inverse of $11$ is $19$ since $11+19=0$.

I know of theory that $\Bbb Z_n$ it's an abelian group but I do not understand why it is.

Recall that an abelian group is a group $G$ where $g\circ h=h\circ g$ for all $g,h\in G$ for some binary operation $\circ$. So, in the case of $\Bbb Z_{30 }$ say, does the relation $g+h=h+g$ hold for all $g,h\in\Bbb Z_{30}$?

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Do feel free to let me know if this is still confusing or you have any further queries.

Answer 2

I suggest you start by writing out the addition tables for $\mathbb{Z}_n$ for some small values of $n$. See that the group axioms are satisfied. (Checking associativity for all the combinations would be tedious - just do a few.) Then you will have a better understanding of those groups, and of the proof that they are groups. Then find all the subgroups of those groups, by working with the tables. You should be able to do this reasonably quickly for all $n \le 8$. When you're done you'll be able to answer your question about $\mathbb{Z}_{30}$ in one or two lines.

In general, the way to understand theorems is by understanding many examples. See Is it always possible to create a intuition for abstract algebra theorems?

Answer 3

For the first part of you question, notice that $S$ contains only three elements: you can build the addition table for $S$ and verify that $S$ is closed with respect to it:

$0$ is the neutral element,

$10+10=20$,

$10+20=0$,

$20+20=10$.