ining ▁the ▁B etti ▁numbers ▁of ▁compact ▁Lie ▁groups , ▁again ▁reducing ▁the ▁problem ▁to ▁an ▁algebraic ▁question ▁on ▁their ▁Lie ▁al gebras , ▁which ▁has ▁since ▁been ▁completely ▁solved . ▁ ▁Lie ▁pseud og r oups ▁After ▁solving ▁the ▁problem ▁of ▁the ▁structure ▁of ▁Lie ▁groups ▁which ▁Cart an ▁( follow ing ▁Lie ) ▁called ▁" finite ▁continuous ▁groups " ▁( or ▁" finite ▁transformation ▁groups "), ▁Cart an ▁pos ed ▁the ▁similar ▁problem ▁for ▁" in finite ▁continuous ▁groups ", ▁which ▁are ▁now ▁called ▁Lie ▁pseud og r oups , ▁an ▁infinite - dimensional ▁analog ue ▁of ▁Lie ▁groups ▁( there ▁are ▁other ▁infinite ▁general izations ▁of ▁Lie ▁groups ). ▁The ▁Lie ▁pseud og r oup ▁considered ▁by ▁Cart an ▁is ▁a ▁set ▁of ▁transformations ▁between ▁subsets ▁of ▁a ▁space ▁that ▁contains ▁the ▁identical ▁transformation ▁and ▁poss esses ▁the ▁property ▁that ▁the ▁result ▁of ▁composition ▁of ▁two ▁transformations ▁in ▁this ▁set ▁( when ever ▁this ▁is ▁possible ) ▁belongs ▁to ▁the ▁same ▁set . ▁Since ▁the ▁composition ▁of ▁two ▁transformations ▁is ▁not ▁always ▁possible , ▁the ▁set ▁of ▁transformations ▁is ▁not ▁a ▁group ▁( but ▁a ▁gr ou po id ▁in ▁modern ▁termin ology ), ▁thus ▁the ▁name ▁pseud og r oup . ▁Cart an ▁considered ▁only ▁those ▁transformations ▁of ▁manif olds ▁for ▁which ▁there ▁is ▁no ▁sub div ision ▁of ▁manif olds ▁into ▁the ▁classes ▁trans posed ▁by ▁the ▁transformations ▁under ▁consideration . ▁Such ▁pseud og r oups ▁of ▁transformations ▁are ▁called ▁primitive . ▁Cart an ▁showed ▁that ▁every ▁infinite