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Math Help - find a noncyclic group

  1. #1
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    Question find a noncyclic group

    Find noncyclic group of order 4 in U(40)
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  2. #2
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    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.
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  3. #3
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    Quote Originally Posted by gosualite View Post
    --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.
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    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.
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  5. #5
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    Quote Originally Posted by gosualite View Post
    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.
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