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