1. The problem statement, all variables and given/known data
Let [itex]A = A(p)\times A'[/itex] where [itex]A(p)[/itex] is a finite commutative p-group (i.e the group has order [itex]p^a[/itex] for [itex]p[/itex] prime and [itex]a>0[/itex]) and [itex]A'[/itex] is a finite commutative group whose order is not divisible by [itex]p[/itex].
Prove that all elements of [itex]A[/itex] of orders [itex]p^k, k\geq0[/itex] belong to [itex]A(p)[/itex]
3. The attempt at a solution
I dont know where to begin with this. Im quite sure that if the order of [itex]A'[/itex] is not divisible by [itex]p[/itex] then the order of any element of [itex]A'[/itex] is not divisible by [itex]p^k[/itex]. Is this usefull or not?
Let [itex]A = A(p)\times A'[/itex] where [itex]A(p)[/itex] is a finite commutative p-group (i.e the group has order [itex]p^a[/itex] for [itex]p[/itex] prime and [itex]a>0[/itex]) and [itex]A'[/itex] is a finite commutative group whose order is not divisible by [itex]p[/itex].
Prove that all elements of [itex]A[/itex] of orders [itex]p^k, k\geq0[/itex] belong to [itex]A(p)[/itex]
3. The attempt at a solution
I dont know where to begin with this. Im quite sure that if the order of [itex]A'[/itex] is not divisible by [itex]p[/itex] then the order of any element of [itex]A'[/itex] is not divisible by [itex]p^k[/itex]. Is this usefull or not?
0 commentaires:
Enregistrer un commentaire