Subgroup of given order of an Abelian group

[siddhuiitb]

Hey!
We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse?
Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not hold in general for Non-Abelian groups. But what about Abelian groups?

 

[Hurkyl]

Suppose that were true. what if m = |G|?

 

[lavinia]

Hey!
We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse?
Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not hold in general for Non-Abelian groups. But what about Abelian groups?

Take a look at an elementary Abelian 2 -group of order greater than 2 e.g. Z/2 x Z/2 x Z/2.

 

[siddhuiitb]

Thanks!