Abstract Algebra: Solving Stumping Questions | αη = β and G is Abelian

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SUMMARY

The discussion focuses on two algebra problems involving permutations and group theory. The first problem requires determining η in the equation αη = β, with α and β defined as specific permutations in S17. The second problem proves that a group G, where a2 = 1 for all a ∈ G, is Abelian, using the hint to analyze (ab)2. The user successfully solved the second problem and proposed a solution for the first, but sought confirmation on the correctness of η.

PREREQUISITES
  • Understanding of permutation groups, specifically Sn notation.
  • Familiarity with group theory concepts, particularly Abelian groups.
  • Knowledge of cycle notation for permutations.
  • Basic algebraic manipulation skills to handle equations involving permutations.
NEXT STEPS
  • Study the properties of permutations in symmetric groups, focusing on cycle decomposition.
  • Learn about group homomorphisms and their implications for group structure.
  • Explore the concept of conjugacy in groups and its relation to Abelian properties.
  • Practice solving problems involving disjoint cycles and their applications in algebra.
USEFUL FOR

Students and educators in abstract algebra, mathematicians interested in group theory, and anyone seeking to deepen their understanding of permutations and their properties.

m-chan
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I have 2 algebra questions which are stumping me, I just can't seem to use my notes to figure them out!

1. Let α, β ∈ S17 where α = (17 2)(1 2 15 17 ), β = (2 3 16)(6 16 17 ).
Determine η, as a product of disjoint cycles, where αη = β.

2. Let G be a group in which a^2 = 1 for all a ∈ G. Prove that G is Abelian.
Hint: Consider (ab)^2.

HELP PLEASE :(
 
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For 2, consider what (ab)2 equals.
 
Right, I've figured out 2, thanks Mark44 and I've done some of 1, but I'm stuck at the end of the question.

I have η= (2 17)(17 15 2 1)(2 3 16)(6 16 17), but I'm not sure if that's right though. I also don't know where to go from there.
 

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