This is my first course in abstract algebra and so far I am only learning about groups. So is there anyone who can explain to me why Klein 4-group is so special that it warrants a category of its own. Please explain as clearly as possible as I am still new to algebra. Thanks
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$\begingroup$The smallest anything is often of interest. One reason is that it lets us think about why it works when nothing smaller does. In this case, it hints that prime groups are always cyclic, and might provide a motivation to prove Lagrange's theorem, which is extremely useful.
$\endgroup$ $\begingroup$It's simply $\mathbb{Z}_2\times \mathbb{Z}_2$, which ordinarily wouldn't merit a special name; the alternative name for it is just a historical artifact. Klein wrote about it in 1884, though, when what would be recognizable as modern group theory was in its infancy. Cayley's wrote about groups in the 1950s; permutation groups popped around the 1870s (though Galois' work preceded it by about 40 years); Lie wrote about Lie groups in the 1880s; and so on.
$\endgroup$ $\begingroup$It's the symmetry group of the rectangle. Don't you think rectangles are important enough to warrant a category of their own?
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