Generate All Permutations of Sn from An and 1 Odd Permutation

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SUMMARY

The discussion confirms that given the alternating group An and one odd permutation, it is possible to generate the entire symmetric group Sn for any n ≥ 2. The user tested this with S3, successfully generating all permutations by multiplying even permutations in A3 with an odd permutation. The principle of group multiplication being a bijection is highlighted, specifically using the equation ax=b, which leads to the conclusion that this property holds for all n. The case for n=1 is noted as a special scenario where A1 equals S1.

PREREQUISITES
  • Understanding of permutation groups, specifically Sn and An.
  • Familiarity with group theory concepts, including bijections and group multiplication.
  • Knowledge of odd and even permutations within symmetric groups.
  • Basic algebraic manipulation involving equations of the form ax=b.
NEXT STEPS
  • Study the properties of symmetric groups, focusing on Sn and An.
  • Explore the concept of odd and even permutations in detail.
  • Learn about group theory bijections and their implications in group operations.
  • Investigate the implications of generating sets in algebraic structures.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in group theory and permutation groups will benefit from this discussion.

gravenewworld
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say you have the alternating group An for some permutation group Sn. If you are given An and then 1 odd permutation, must you be able to generate all of Sn? I tried it for S3 and I multiplied all the even perms in S3 by only 1 element that wasn't in A3 and was able to generate all of S3. Does this hold for any n?
 
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Yes, that is generally true.
Note that in any group multiplication on the left by an element in the group is a bijection.
[tex]ax=b \iff x=a^{-1}b[/tex]
Use this to prove it for the general case [itex]S_n ,n\geq 2[/itex]
The case n=1 is special, since the A1=S1.
 
Alright thanks a lot galileo. I just wanted to be sure of that fact before I brought it up in my presentation that I have to give.
 

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