▁the ▁matrix ▁A ▁only . ▁ ▁Similarly , ▁for ▁the ▁special ▁orthogonal ▁group ▁SO ( n ) ▁we ▁have ▁ ▁SO ( n ) ▁ ▁E +( n ) ▁/ ▁T ▁ ▁In version ▁The ▁conjug ate ▁of ▁the ▁in version ▁in ▁a ▁point ▁by ▁a ▁translation ▁is ▁the ▁in version ▁in ▁the ▁translated ▁point , ▁etc . ▁ ▁Thus ▁the ▁conjug acy ▁class ▁within ▁the ▁E uclidean ▁group ▁E ( n ) ▁of ▁in version ▁in ▁a ▁point ▁is ▁the ▁set ▁of ▁in versions ▁in ▁all ▁points . ▁ ▁Since ▁a ▁combination ▁of ▁two ▁in versions ▁is ▁a ▁translation , ▁the ▁conjug ate ▁closure ▁of ▁a ▁singleton ▁containing ▁ ▁in version ▁in ▁a ▁point ▁is ▁the ▁set ▁of ▁all ▁transl ations ▁and ▁the ▁in versions ▁in ▁all ▁points . ▁This ▁is ▁the ▁generalized ▁di hed ral ▁group ▁di h ▁( R n ). ▁ ▁Similarly ▁{ ▁I , ▁− I ▁} ▁is ▁a ▁normal ▁subgroup ▁of ▁O ( n ), ▁and ▁we ▁have : ▁ ▁E ( n ) ▁/ ▁di h ▁( R n ) ▁ ▁O ( n ) ▁/ ▁{ ▁I , ▁− I ▁} ▁ ▁For ▁odd ▁n ▁we ▁also ▁have : ▁O ( n ) ▁ ▁SO ( n ) ▁× ▁{ ▁I , ▁− I ▁} ▁and ▁hence ▁not ▁only ▁O ( n ) ▁/ ▁SO ( n ) ▁ ▁{ ▁I , ▁− I ▁} ▁but ▁also : ▁O ( n ) ▁/ ▁{ ▁I , ▁− I ▁} ▁ ▁SO ( n ) ▁