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Exercise: Mark True or False. Explain why.

a) The symmetry group of a square is isomorphic to $\mathbb{Z_8}$.

b) The symmetry group of a square is isomorphic to $\mathbb{S_8}$.

c)The symmetry group of a square is isomorphic to a subgroup $\mathbb{S_8}$.

I need explanation with isomorphism. I know that symmetry group of a square is dihedral group $\mathbb{D_4}$ which has 8 elements. $\mathbb{S_8}$ means all permutations. $\mathbb{Z_8}$ cyclic group of order 8.

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closed as off-topic by Derek Holt, Christoph, Shaun, Alexander Gruber May 20 at 18:14

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$D_4$ is not abelian, but $\mathbb{Z}_8$ is. So they are not isomorphic.

$D_4$ has order $8$ but $S_8$ has order $8!$ and so they are not isomorphic.

$D_4$ is a permutation group of a set of cardinality $4$, so $D_4$ is isomorphic to a subgroup of $S_4$, and so a subgroup of $S_8$.

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  • $\begingroup$ Also $D_8$ is a group of order $8$, so it is isomorphic to a subgroup of $S_8$. $\endgroup$ – bof May 20 at 6:53
  • $\begingroup$ Exactly, by Cayley. $\endgroup$ – Hongyi Huang May 20 at 6:54

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