# When are groups finite?

1. Jun 27, 2010

### Identity

Is it true that cyclic groups with $$x^n = 1$$ the only finite groups (with order n)?

I've been experimenting with a few groups and I think this is true but I'm not sure.

thanks

2. Jun 27, 2010

### CompuChip

Have you looked at the permutation groups Sn, dihedral groups Dn (symmetries of regular polygons), etc.?

In fact, almost all finite groups are non-abelian, so the cyclic groups are just very few in number.

3. Jun 27, 2010

### phyzguy

The finite groups are completely categorized. See for example:

http://en.wikipedia.org/wiki/List_of_finite_simple_groups

Or, I strongly recommend Mark Ronan's book, "Symmetry and the Monster". It is a good read of this fascinating subject.

4. Jun 28, 2010

### Martin Rattigan

There are as many cyclic groups ($\aleph_0$) as finite groups. By, "almost all finite groups are non-abelian", do you mean $\lim_{n\rightarrow \infty}a_n/g_n=0$, where $a_n$ is the number of abelian groups of order $\leq n$ and $g_n$ is the number of groups of order $\leq n$? (This doesn't seem likely.)

5. Jun 28, 2010

### HallsofIvy

Given any n, there certainly exists a cylic group of order n.

It is also true that if p is a prime number then then the only group of order p is the cylic group.

But if n is NOT prime, then there exist other, non-cyclic, groups of order n.

The simplest example is for n= 4. The "Klein 4-group" is not cyclic.

The Klein 4-group has 4 members, e, a, b, c satifying
ee= e, ea= a, eb= b, ec= c (e is the identity)
ae= a, aa= e, ab= c, ac= b
be= b, ba= c, bb= e, bc= a
ce= c, ca= b, cb= a, cc= e

The fact that every element is its own inverse proves this group is not cyclic.

Any group of order 4 is isomorphic either to the cyclic group of order 4 or to the Klein four-group.

6. Jun 28, 2010

### Martin Rattigan

Then again it doesn't seem unlikely. Has it been proved?

The non-abelian groups of orders 2n will probably arrange it by themselves (e.g. there are over ten times as many non-abelian groups of order 1024 as there are other groups up to and including 1024) so this may not be too difficult to prove.

Last edited: Jun 28, 2010
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