We know from class that the dihedral group Ds = (r, s) = {1, r, r,...

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We know from class that the dihedral group Ds = (r, s) = {1, r, r², r³, s, sr, sr², sr³} is the group of symmetries of a square, which is generated by a rotation r by 2π/4 and a reflection s. (a) Find all subgroups of Dg order 2. 6 (b) Show that Ds has exactly three subgroups of order 4, one of which is cyclic, while the remaining two are non-cyclic. (Note that this gives an example of a non-abelian group of order 4.) (c) Assuming that isomorphic groups possess the same subgroup struc- ture, establish that Qs is not isomorphic to Ds. We know from class that the dihedral group Ds = (r, s) = {1, r, r², r³, s, sr, sr², sr³} is the group of symmetries of a square, which is generated by a rotation r by 2π/4 and a reflection s. (a) Find all subgroups of Dg order 2. 6 (b) Show that Ds has exactly three subgroups of order 4, one of which is cyclic, while the remaining two are non-cyclic. (Note that this gives an example of a non-abelian group of order 4.) (c) Assuming that isomorphic groups possess the same subgroup struc- ture, establish that Qs is not isomorphic to Ds.

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