▁whole ▁E ( n ). ▁ ▁Rot ore f lection ▁The ▁left ▁and ▁also ▁the ▁right ▁cos et ▁of ▁a ▁reflection ▁in ▁a ▁plane ▁combined ▁with ▁a ▁rotation ▁by ▁a ▁given ▁angle ▁about ▁a ▁per pend icular ▁axis ▁is ▁the ▁set ▁of ▁all ▁combinations ▁of ▁a ▁reflection ▁in ▁the ▁same ▁or ▁a ▁parallel ▁plane , ▁combined ▁with ▁a ▁rotation ▁by ▁the ▁same ▁angle ▁about ▁the ▁same ▁or ▁a ▁parallel ▁axis , ▁pres erving ▁orientation ▁ ▁Is ometry ▁groups ▁Two ▁is ometry ▁groups ▁are ▁said ▁to ▁be ▁equal ▁up ▁to ▁conjug acy ▁with ▁respect ▁to ▁aff ine ▁transformations ▁if ▁there ▁is ▁an ▁aff ine ▁transformation ▁such ▁that ▁all ▁elements ▁of ▁one ▁group ▁are ▁obtained ▁by ▁taking ▁the ▁conjug ates ▁by ▁that ▁aff ine ▁transformation ▁of ▁all ▁elements ▁of ▁the ▁other ▁group . ▁This ▁applies ▁for ▁example ▁for ▁the ▁symmetry ▁groups ▁of ▁two ▁patterns ▁which ▁are ▁both ▁of ▁a ▁particular ▁wall paper ▁group ▁type . ▁If ▁we ▁would ▁just ▁consider ▁conjug acy ▁with ▁respect ▁to ▁is omet ries , ▁we ▁would ▁not ▁allow ▁for ▁scaling , ▁and ▁in ▁the ▁case ▁of ▁a ▁par alle lo gram met ic ▁lattice , ▁change ▁of ▁shape ▁of ▁the ▁par alle lo gram . ▁Note ▁however ▁that ▁the ▁conjug ate ▁with ▁respect ▁to ▁an ▁aff ine ▁transformation ▁of ▁an ▁is ometry ▁is ▁in ▁general ▁not ▁an ▁is ometry , ▁although ▁volume ▁( in ▁ 2 D : ▁area ) ▁and ▁orientation ▁are ▁preserved . ▁ ▁Cy clic ▁groups ▁Cy clic ▁groups ▁are ▁Ab elian , ▁so ▁the ▁conjug ate