Isomorphic Groups: Same Info Studying 1 or Both

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Discussion Overview

The discussion revolves around the concept of isomorphic groups in abstract algebra, specifically whether studying one isomorphic group provides all necessary algebraic information about another. Participants explore the definition and implications of "algebraic properties" in the context of isomorphism.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants propose that studying one isomorphic group allows for the deduction of all algebraic information about the other.
  • Others argue that the definition of "algebraic properties" is crucial to this assertion.
  • A participant questions what might be considered algebraic properties that could differ between isomorphic groups.
  • It is suggested that some definitions of algebraic properties include only those that are preserved under isomorphism.

Areas of Agreement / Disagreement

Participants generally agree that the definition of algebraic properties is significant, but there is no consensus on what constitutes these properties or whether they can differ in isomorphic groups.

Contextual Notes

Limitations in the discussion include the ambiguity surrounding the definition of "algebraic properties" and the potential for differing interpretations among participants.

tgt
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If two groups A and B are isomorphic then by studying one of them, we can deduce all algebraic information about the other? Hence studying one is equivalent to studying the other?
 
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Provided you define "algebraic properties" correctly, then yes.
 
quasar987 said:
Provided you define "algebraic properties" correctly, then yes.

Isn't algebraic properties 'clear cut'? What are some things that might be considered algebraic properties but are different in two isomorphic groups?

Would some even define algebraic properties to be those that occur in all isomorphic groups?
 
tgt said:
Would some even define algebraic properties to be those that occur in all isomorphic groups?
Yes: usually an algebraic property is defined as a property which is preserved under isomorphism.
 

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