▁by ▁every ▁element ▁of ▁every ▁element ▁is ▁the ▁latter . ▁ ▁Z mn ▁/ ▁Z m ▁ ▁Z n . ▁ ▁Z mn ▁is ▁the ▁direct ▁product ▁of ▁Z m ▁and ▁Z n ▁if ▁and ▁only ▁if ▁m ▁and ▁n ▁are ▁cop r ime . ▁Thus ▁e . g . ▁Z 1 2 ▁is ▁the ▁direct ▁product ▁of ▁Z 3 ▁and ▁Z 4 , ▁but ▁not ▁of ▁Z 6 ▁and ▁Z 2 . ▁ ▁Di hed ral ▁groups ▁Consider ▁the ▁ 2 D ▁is ometry ▁point ▁group ▁D n . ▁The ▁conjug ates ▁of ▁a ▁rotation ▁are ▁the ▁same ▁as ▁the ▁inverse ▁rotation . ▁The ▁conjug ates ▁of ▁a ▁reflection ▁are ▁the ▁ref lections ▁rot ated ▁by ▁any ▁multiple ▁of ▁the ▁full ▁rotation ▁unit . ▁For ▁odd ▁n ▁these ▁are ▁all ▁ref lections , ▁for ▁even ▁n ▁half ▁of ▁them . ▁ ▁This ▁group , ▁and ▁more ▁generally , ▁abstract ▁group ▁Di hn , ▁has ▁the ▁normal ▁subgroup ▁Z m ▁for ▁all ▁divis ors ▁m ▁of ▁n , ▁including ▁n ▁itself . ▁ ▁Additionally , ▁Di h 2 n ▁has ▁two ▁normal ▁sub groups ▁is omorphic ▁with ▁Di hn . ▁They ▁both ▁contain ▁the ▁same ▁group ▁elements ▁forming ▁the ▁group ▁Z n , ▁but ▁each ▁has ▁addition ally ▁one ▁of ▁the ▁two ▁conjug acy ▁classes ▁of ▁Di h 2 n ▁\ ▁Z 2 n . ▁ ▁In ▁fact : ▁Di hm n ▁/ ▁Z n ▁ ▁Di hn ▁Di h 2 n ▁/ ▁Di hn ▁ ▁Z 2 ▁Di h 4 n +