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How to find subgroup of index n in a given group

  1. Apr 6, 2012 #1
    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.
  2. jcsd
  3. Apr 6, 2012 #2

    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.

  4. Apr 6, 2012 #3


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    I'm not an expert on these sorts of things, but I just did a literature search and managed to find the following relevant article:

    M. Conder and P. Dobcsányi, Applications and adaptations of the low index subgroups procedure, Math. Comp. 74 (2005), 485-497.

    The first couple of paragraphs read:
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