# Thread: find a noncyclic group

1. ## find a noncyclic group

Find noncyclic group of order 4 in U(40)

2. I have no idea what U(40) is, but there is only one noncyclic group of order 4 (up to isomorphism). Take a gander at the multiplicative group $\mathbb{Z}_8^\times$, count its elements and see if it's cyclic.

--Or look up a wonderful creature called the Klein 4-group.

3. Originally Posted by gosualite
--Or look up a wonderful creature called the Klein 4-group.
There is nothing wonderful about it. I am supprised why such a basic and uninteresting group would be named after somebody. I can understand naming general linear or projective groups after people but why this? Do you happen know why? That is on thing that bothered me.

4. It's a great example for a lot of things. Like if you need a random abelian noncyclic group, look no further. And hello noncyclic p-group. It's also nice that every element has order 2 so you can attach it to other groups and easily visualize what's going on. A lot of problems that arise in the study of abelian group structure can be simplified by working out the case for Z2+Z2 too because a copy of it lives inside every noncyclic abelian group of even order. I stand by my characterization of the Klein 4 as a wonderful creature.

5. Originally Posted by gosualite
It's a great example for a lot of things. Like if you need a random abelian noncyclic group, look no further. And hello noncyclic p-group. It's also nice that every element has order 2 so you can attach it to other groups and easily visualize what's going on. A lot of problems that arise in the study of abelian group structure can be simplified by working out the case for Z2+Z2 too because a copy of it lives inside every noncyclic abelian group of even order. I stand by my characterization of the Klein 4 as a wonderful creature.
I guess you like to use it as a counterexample.
Still it seems I cannot develope Kleiness appreciation.