# How to find subgroup of index n in a given group

Dear Folks:
Is there a general method to find all subgroups in a given abstract group?? Many Thanks!!

This question came into my classmates' mind when he wants to find a 2 sheet covering of the Klein Bottle. This question is equivalent to find a subgroup of index 2 in Z free product Z/2Z. When n = 2, this question is sovalble using homology method according to my tutor, since the subgroup of index 2 is always regular, but when it comes to the general case, he says he suspected it has been solved.

PS: I'm sorry for my poor English. (I'm not a native speaker.) Hope I've expained it clearly.

Dear Folks:
Is there a general method to find all subgroups in a given abstract group?? Many Thanks!!

This question came into my classmates' mind when he wants to find a 2 sheet covering of the Klein Bottle. This question is equivalent to find a subgroup of index 2 in Z free product Z/2Z. When n = 2, this question is sovalble using homology method according to my tutor, since the subgroup of index 2 is always regular, but when it comes to the general case, he says he suspected it has been solved.

PS: I'm sorry for my poor English. (I'm not a native speaker.) Hope I've expained it clearly.

I think your instructor is right: in general is, as far as I know, impossible to decide what orders and indexes subgroups of a group can have.

Of course, in particular cases we can: a finite abelian group ALWAYS has a subgroup of order (index) d, for any divisor d of the group's order. But there is hardly something more general than this, I'm afraid.

DonAntonio

morphism