Can $S_4$ (symmetry group of $4$) be represented by the union of $D_4$ (dihedral group of $4$) and the cosets (in $S_4$) thereof? If not why not?
S4 decompose into D4 and cosets?
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group-theory
symmetric-groups
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0Every group is equal the union of a subgroup and the cosets of the subgroup. Whether by $D_4$ you mean the dihedral group of order 4 or of order 8, both are subgroups of $S_4$, so $S_4$ is the union of the cosets of this subgroup. – 2011-05-09
2 Answers
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You have effectively answered your own question by talking about "the cosets of $D_4$ in $S_4$". This only makes sense because $D_4$ can be regarded as a subgroup of $S_4$ (namely the subgroup of permutations of the corners of a square that can be obtained by rotations and reflections). For every subgroup of a group, the group is the union of the subgroup and its cosets, so the answer is "yes".
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Presumably, "dihedral group of 4" means the group of symmetries of a square (this is often denoted $D_8$ since it has 8 elements). If so, then answer is Yes - it's a subgroup of $S_4$ and, therefore, its cosets cover the whole group.
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0Rather than say $D_8$ is a subgroup of $S_4$, I'd say $S_4$ has a subgroup isomorphic to $D_8$. – 2011-05-09