Isomorphic Groups: Same Info Studying 1 or Both

[tgt]

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?

 

[quasar987]

Provided you define "algebraic properties" correctly, then yes.

 

[tgt]

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?

 

[morphism]

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.