Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.
Every cyClic grouP oF Prime oRder is A sImple grOUp, whIch caNnot bE brokEN down inTo sMalLer groUps. in the cLasSification Of FiniTe simPLE grOups, ONe oF the three InFINiTe claSsEs cOnsiSts Of the CYclIc gRoUps of pRime oRdEr. THe CYCLiC gRoupS of PrIme oRdeR ARe thus amONg the BuildiNG blocks From WhicH all GroupS can be BuiLt.