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

  • Context: Graduate 
  • Thread starter Thread starter clkt
  • Start date Start date
  • Tags Tags
    Isomorphism
Click For Summary
SUMMARY

The discussion focuses on finding an isomorphism between a subgroup of the symmetric group Sn+m and the direct product Zn x Zm. It is established that Sn+m is generally not isomorphic to Zn x Zm, but specific cases can be analyzed, such as when n and m are both equal to 2. The key takeaway is that if two subgroups H and K of a group G have a trivial intersection and commute, then the subgroup HK is isomorphic to H x K.

PREREQUISITES
  • Understanding of group theory concepts, specifically symmetric groups.
  • Familiarity with subgroup properties and isomorphisms.
  • Knowledge of direct products in algebra.
  • Basic experience with mathematical notation and terminology.
NEXT STEPS
  • Study the properties of symmetric groups, particularly Sn+m.
  • Learn about subgroup structures and their isomorphisms in group theory.
  • Explore the concept of trivial intersections and commutativity in groups.
  • Investigate specific cases of isomorphisms between subgroups of Sn+m and direct products.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, as well as students studying group theory and its applications.

clkt
Messages
6
Reaction score
0
How do I find an isomorphism between Sn+m and Zn x Zm? provided n,m are not relatively prime? Thanks.
 
Physics news on Phys.org
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!
 

Similar threads

Undergrad ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
  • · Replies 31 ·
2
Replies
31
Views
2K
Graduate Isomorphism concepts,( example periods elliptic functions )
  • · Replies 8 ·
Replies
8
Views
2K
MHB Isomorphism between $M$ and $S_3$: Finding the Corresponding Function
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad Meaning of "identify" & variations in different math context
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Looking for a creative or quick method for finding roots in GF(p^n)
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad A regular matrix <=> mA isomorphism
  • · Replies 11 ·
Replies
11
Views
3K
Graduate Examine Structurally Distinct Abelian Groups with Primary Decomposition Thm
  • · Replies 3 ·
Replies
3
Views
2K
MHB Prove that CnXCm isomorphic to Cgcd(m,n)XClcm(m,n)
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Question about normal subgroups/Lattice Isomorphism Theorem
  • · Replies 1 ·
Replies
1
Views
2K
How many mutually nonisomorphic Abelian groups of order 50
  • · Replies 2 ·
Replies
2
Views
3K