How to find subgroup of index n in a given group

  • #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.
 

Answers and Replies

  • #2
606
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.



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
 
  • #3
morphism
Science Advisor
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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:
Given a finitely presented group ##G = \langle X \mid R \rangle##, where X is a finite set of generators and R is a finite set of relators (each expressed as a word on ##X\cup X^{-1}##), it is frequently desirable to find all subgroups of G of index up to some specified integer N, or at least a representative of each conjugacy class of subgroups of index up to N.

A complete enumeration of such subgroups is algorithmically feasible for two reasons. First, there are finitely many subgroups of up to a given index in any finitely generated group G, since every subgroup H of index n corresponds to a homomorphism ##G \to S_n## (equivalent to the representation of G on right cosets of H), and there are only finitely many such homomorphisms (since there are only finitely many possibilities for the image of each of the elements from a finitely generating set). Second, by Schreier's theorem, every subgroup of nite index in a finitely generated group is itself finitely generated.
 

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