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The dihedral group D6-geometrically, the group of symmetries of a regular hexagoncan be realized as...

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The dihedral group D6-geometrically, the group of symmetries of a regular hexagoncan be realized as a subgroup of S6 by enumerating the vertices (see figure below): The symmetry shown corresponds to reflection around the dotted altitude. Looking at the action on the numbered vertices, we identify this in S6 as (12)(36)(45). With this identification, we have D6={e,(123456),(135)(246),(14)(25)(36),(153)(264),(165432)(12)(36)(45),(16)(25)(34),(14)(23)(56),(26)(35),(13)(46),(15)(24)} (a) Find a pair of elements in D6 which do not commute. (b) Calculate the orders of the elements of D6. Is D6A4 ? (c) Show that D6 is solvable. The dihedral group D6-geometrically, the group of symmetries of a regular hexagoncan be realized as a subgroup of S6 by enumerating the vertices (see figure below): The symmetry shown corresponds to reflection around the dotted altitude. Looking at the action on the numbered vertices, we identify this in S6 as (12)(36)(45). With this identification, we have D6={e,(123456),(135)(246),(14)(25)(36),(153)(264),(165432)(12)(36)(45),(16)(25)(34),(14)(23)(56),(26)(35),(13)(46),(15)(24)} (a) Find a pair of elements in D6 which do not commute. (b) Calculate the orders of the elements of D6. Is D6A4 ? (c) Show that D6 is solvable

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