Need to show why $(S_7,\circ)$ is not isomorphic to $(\Bbb Z/100\Bbb Z,+)$. I think it might have something to do with Abelian, but I'm not sure.
2026-04-03 23:16:36.1775258196
Why not isomorphic?
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In my opinion, the simplest way to show that $S_7$ is not isomorphic to $\Bbb{Z}/100\Bbb{Z}$ is to consider their cardinalities.
$S_7$ has $7! = 5040$ elements, and $\Bbb{Z}/100\Bbb{Z}$ has $100$ elements. Since the two groups have different cardinalities, there cannot exist a bijection between the two. An isomorphism is necessarily a bijection, so there cannot exist an isomorphism between $S_7$ and $\Bbb{Z}/100\Bbb{Z}$. Hence, the two are not isomorphic.