What is the order of Klein 4 group?

The Wikipedia said it is isomorphic to the Dihedral group of order 4.

But isomorphism preserves group order hence Klein 4 group should have group order 4.

But I couldn't find any elements in Klein 4 group has group order 4.( Since there are only 4 elements and each non-identity element is self inverse and thus has order of 2 )

Can anyone tell me where is going wrong?

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    $\begingroup$ I think you're confusing two things: the order of a group, and the order of an element. They are different $\endgroup$ – D_S Dec 14 '19 at 4:45
  • $\begingroup$ @rschwieb: Wikipedia does say the Klein group is isomorphic to the dihedral group of order four $\endgroup$ – J. W. Tanner Dec 15 '19 at 4:39
  • $\begingroup$ @J.W.Tanner igh: I know what happened: page search on mobile didn’t search collapsed tabs. $\endgroup$ – rschwieb Dec 15 '19 at 13:02
  • $\begingroup$ I have never heard the presentation with $n=2$ included. I had thought the whole point was symmetries of an $n$ gon, but it looks like the wiki article throws in that exceptional case. $\endgroup$ – rschwieb Dec 15 '19 at 13:06

The Klein group is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$ (I am here using the convention that $\mathbb{Z}/n\mathbb{Z}=\mathbb{Z}_n$ for $n \in \mathbb{N}$), so it is indeed of order $4$ not having however any element of order $4$ (all its nontrivial elements are of order $2$). You seem to be harbouring the expectation that if a finite group $G$ is of order $n$ then it will admit a certain element of order $n$, in other words that it will be cyclic, generated by that respective element; that of course very rarely is the case, as in only very particular situations will a finite group be cyclic!! The majority of them are rather not!!

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  • $\begingroup$ Thank you. So in general how to find the group order of a group since it is not helpful to find the order of each element? $\endgroup$ – 张耀威 Dec 14 '19 at 4:55
  • $\begingroup$ Ok never mind. I think I know it now. Just count the cardinality. $\endgroup$ – 张耀威 Dec 14 '19 at 4:58
  • $\begingroup$ By definition, the order of a group is the cardinality of its support set (also known as underlying set, in English terminology), so indeed in order to find the order of a group you need to evaluate the cardinality of a certain set. $\endgroup$ – ΑΘΩ Dec 14 '19 at 6:34

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