# Thread: finding generator of a cyclic group

1. ## finding generator of a cyclic group

How do i find the generator for this cyclic group? Im thinking about <5m+7n> but im not sure if its correct. Would gladly like to know if there are any methods to find the generator of such cyclic group. On a side note i guess it should be m,n in Z instead of m.n

2. ## Re: finding generator of a cyclic group

Originally Posted by noobpronoobpro

How do i find the generator for this cyclic group? Im thinking about <5m+7n> but im not sure if its correct. Would gladly like to know if there are any methods to find the generator of such cyclic group. On a side note i guess it should be m,n in Z instead of m.n
Has it occurred to you that the question gives the set underling the group, but nothing about the group operation? Now a reasonable guess is that the group operation is simple addition. Is that correct?

In any case, that group contains $\{0,\pm 5, \pm7,\pm 35,\pm 36,\pm 36,\pm 37,\cdots\}$ can you show that?

Now assume that addition, $\oplus$ is the operation.
1) What is the identity of the operation ?
2) If $t$ is an element is that set, what is the operational inverse of $t~?$.
3) If $t~\&~s$ are elements is that set, can you show that $(-t)\oplus s$ is in the set?

Having done 1) & 2), if you can do 3) then you have proved that is a group(subgroup).

Can you find a generator?

3. ## Re: finding generator of a cyclic group

im confused with how you arrived at the various elements, can you give me some help?

4. ## Re: finding generator of a cyclic group

If you are referring to "{0,±5,±7,±35,±36,±36,±37,⋯}", those are just 5m+ 7n with m= n= 0, then m= ±1, n= 0, m= 0, n= ±1, etc. (Having "±36" twice is a typo of course).
Is it true that the group operation is addition? What is the definition of "generator" of a group?

5. ## Re: finding generator of a cyclic group

A generator is able to produce the whole set of elements in a group upon repeated application on itself. So, it doesnt matter if the generator produces elements that are not included in the group? The generator i'm referring to are 1 and -1.

6. ## Re: finding generator of a cyclic group

$\displaystyle H=Z$

since

$\displaystyle x=5(3x)+7(-2x)$

7. ## Re: finding generator of a cyclic group

i see... thats really helpful in solving my queries :>