How do I find an isomorphism between Sn+m and Zn x Zm?

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Discussion Overview

The discussion revolves around finding an isomorphism between a subgroup of the symmetric group Sn+m and the direct product of cyclic groups Zn and Zm, particularly under the condition that n and m are not relatively prime. The scope includes theoretical exploration and clarification of group properties.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant initially inquires about finding an isomorphism between Sn+m and Zn x Zm, questioning the relationship between these groups when n and m are not relatively prime.
  • Another participant humorously suggests unconventional methods for finding an isomorphism and points out that Sn+m is generally not isomorphic to Zn x Zm in standard notation.
  • A later reply clarifies that the original question should focus on finding an isomorphism between a subgroup of Sn+m and Zn x Zm, indicating that specific cases (like n=m=2) can be resolved but a general form remains uncertain.
  • One participant mentions a condition under which the subgroup HK is isomorphic to the direct product HxK, provided certain properties about the subgroups H and K hold.
  • A final response indicates that the inquiring participant believes they have resolved their question, suggesting some progress in their understanding.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original question regarding the isomorphism, and multiple viewpoints and clarifications are presented throughout the discussion.

Contextual Notes

There are limitations regarding the definitions of the groups involved and the specific conditions under which the isomorphism might hold, as well as unresolved aspects of the general form for the isomorphism.

Who May Find This Useful

Readers interested in group theory, particularly those exploring isomorphisms and subgroup properties within symmetric groups and cyclic groups.

clkt
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How do I find an isomorphism between Sn+m and Zn x Zm? provided n,m are not relatively prime? Thanks.
 
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I dunno, place an ad in the classifieds, put up posters, etc. Or maybe start by defining what you even mean by Sn+m, because in the standard notation, Sn+m is a group that's generally not isomorphic to Zn x Zm. Also, post this in the homework section, follow the guidelines for posting homework, and when you start a new thread in the homework section, use the template that will be given to you.
 
Hi, this isn't exactly a homework question... Anyways, rereading my question again, I made a mistake. It should be how do I find an isomorphism between A SUBGROUP in Sn+m and Zn x Zm. If it is specific cases like n,m = 2,2 I can figure it out. But how do I figure out a general form for the isomorphism?
 
If there are two subgroups H and K of a group G such that the intersection of H and K is trivial and hk=kh for all h in H, k in K, then the subgroup HK is isomorphic to HxK.
 
ah, thanks, I think I have it figured out now!
 

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