: Let ( G ) act on set ( S ). Prove if ( G ) acts transitively on ( S ), then for any ( x \in S ), ( |S| = [G : \textStab(x)] ).
Most Sylow problems are "counting games." Use the congruence and the fact that must divide the index to narrow down the possibilities. dummit foote solutions chapter 4
: Every group of order ( p^2 ) is abelian. Solution idea : From 4.3.6, ( |Z(G)| = p ) or ( p^2 ). If ( |Z(G)| = p ), then ( G/Z(G) ) cyclic ⇒ ( G ) abelian (contradiction unless ( Z(G) = G )). : Let ( G ) act on set ( S )
: Let ( G ) act on set ( S ). Prove if ( G ) acts transitively on ( S ), then for any ( x \in S ), ( |S| = [G : \textStab(x)] ).
Most Sylow problems are "counting games." Use the congruence and the fact that must divide the index to narrow down the possibilities.
: Every group of order ( p^2 ) is abelian. Solution idea : From 4.3.6, ( |Z(G)| = p ) or ( p^2 ). If ( |Z(G)| = p ), then ( G/Z(G) ) cyclic ⇒ ( G ) abelian (contradiction unless ( Z(G) = G )).