Groups whose elements have order 2

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SUMMARY

In the discussion, participants analyze the properties of a group G where every non-identity element has order two, concluding that G is commutative. The proof involves demonstrating that for any two elements a and b in G, the equation ab = ba holds true. The proof relies on the associative property of multiplication and the definition of order in group theory.

PREREQUISITES
  • Understanding of group theory concepts, specifically group order
  • Familiarity with the properties of commutative groups
  • Knowledge of associative multiplication in algebraic structures
  • Basic experience with mathematical proofs and logical reasoning
NEXT STEPS
  • Study the properties of groups with elements of finite order
  • Learn about Abelian groups and their characteristics
  • Explore examples of groups where every element has order two, such as Klein four-group
  • Investigate the implications of group commutativity in algebraic structures
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, group theorists, and educators looking to deepen their understanding of group properties and commutativity.

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Homework Statement



suppose that G is a group in which every non-identity element has order two. show that G is commutative.



Homework Equations





The Attempt at a Solution


IS THIS CORRECT?

ab=a[(ab)^2]b=(a^2)(ba)(b^2)=ba
 
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yeah looks good (same as last post), if you want to include every step
ab = (ae)b = aeb = a (abab) b = (aa) ba (bb) = ebae = ba
does assume the multiplication is associative
 
halvizo1031 said:

Homework Statement



suppose that G is a group in which every non-identity element has order two. show that G is commutative.



Homework Equations





The Attempt at a Solution


IS THIS CORRECT?

ab=a[(ab)^2]b=(a^2)(ba)(b^2)=ba

Will you please stop posting the same question over and over again?
 

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