I am having troubles proving this:

Prove that any group on five elements is cyclic.

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- Jan 21st 2010, 05:02 PMHavenGroup on Five Elements
I am having troubles proving this:

Prove that any group on five elements is cyclic. - Jan 21st 2010, 05:07 PMDrexel28
Do you know Lagrange's theorem?

**Lemma:**If a set has no proper non-trivial subgroups, then it's cyclic.

**Proof:**Let meet the criterion. Let be arbitrary. We must then have that is a subgroup of that is non-trivial. Thus, by assumption we must have that is improper, thus equal to . The conclusion follows.

Now, using this lemma and Lagrange's theorem we see that any subgroup of where for some prime must be a divisor of . Thus, any subgroup must be of order . Therefore, the lemma implies that is cyclic for any prime-ordered group.

Your example is just a corollary. - Jan 21st 2010, 08:18 PMHaven
I'm supposed to prove the statement by arguing that all the non-identity elements of G have order 5.

So far I have

Let such that .

so

if , then we are done, since .

Case 1

Then and where .

If

If respectively

If

Since they are all contradictory, x cannot have order 4. So no element in G can have order 4.

But I can't figure out the other cases - Jan 21st 2010, 08:20 PMDrexel28
- Mar 20th 2010, 06:05 AMelements
Fascinating. I was looking up the Taoist 5 Elements and I found your thread about mathematical 5 elements.

Not being a mathematician, I don't get it. Would you mind giving me the basic premise of the mathematical 5 elements concept?

In Taoism, any five cyclical system is represented by wood, fire, earth, metal, and water. How each supports each other is obvious - back down to water, the fifth element supporting wood, the first element all over again. This characterization is supposed to be a perfect analogy for anything set of five that are mutually supportive. I found a fuller description of the 5 elements - Mar 20th 2010, 06:40 AMproscientia