Let H and K be subgroups of the group G, and let S be the set of left oosets of K t Dene agroup action ofH on S by setting a '(IHJ = nxKJora l a E H and x e G. H X By considering the orbit of K under this action, show that IE K I = . Ans : Let G be groupS be a set then a mapis called group action on S if : G S S defined by ( g , s ) gs such that ( i ) es=s , s S ( ii ) ( gh ) s=g ( hs ) , g , h G Now comes the question : Given that The Subgroup H act onthe set S where set S is defined by . S= { gK | g G }, Sets of all of K G Then Action isdefined as ( a , xK ) axK , for all a Hx G . Orbit of K isO ( K )=O ( eK )={ hK|h H } Then automaticaly h H ( hK )=HK Now Since |hK |=|K|, for all h H These Cosets are pairwise disjoint Hence|HK|=|K||O ( K )|... ... . ... ... ... ... .(1) Also We know that |O ( K )|=[ H : H K ]=Index of H K H Where H K = {h H| hK=K } { h H|h K } HK There fore|O ( K )|= |H| = |H| |H K| |H K| ( 1 )( 2 ) we get |H| |HK|=|K| |H K| >|HK |= | K||H| , Proved . |H K| ... ... ... ... ... ... . (2 ) Ans : Let G be groupS be a set then a mapis called group action on S if : G S S defined by ( g , s ) gs such that ( i ) es=s , s S ( ii ) ( gh ) s=g ( hs ) , g , h G Now comes the question : Given that The Subgroup H act onthe set S where set S is defined by . S= { gK | g G }, Sets of all of K G Then Action isdefined as ( a , xK ) axK , for all a Hx G . Orbit of K isO ( K )=O ( eK )={ hK|h H } Then automaticaly h H ( hK )=HK Now Since |hK |=|K|, for all h H These Cosets are pairwise disjoint Hence|HK|=|K||O ( K )|... ... . ... ... ... ... .(1) Also We know that |O ( K )|=[ H : H K ]=Index of H K H Where H K = {h H| hK=K } { h H|h K } HK There fore|O ( K )|= |H| = |H| |H K| |H K| ( 1 )( 2 ) we get |H| |HK|=|K| |H K| >|HK |= | K||H| , Proved . |H K| ... ... ... ... ... ... . (2 )