if a group of order 2p ( p prime) is abelian....then does it have exactly one element of order 2 ?? if a group is non abelian...i could figure out that there are p elements of order 2. but the abelian case is a bit confusing... also..is it like...any group of order 2p has an element of order p? if a group has orer p^a , a>=1 where p is prime....then i've gotta show that G has an element of order p. can i say that any non-identity element in G can have order p or p^2 or p^3......or p^a. then if x in G is of the form x^(p^i) =e ....we can say, (x^(p^(i-1))^p= e and we've found an element x^(p^(i-1)) that is of order p....it seems too simple to be correct.