Isomorphic Groups: Same Info Studying 1 or Both

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In summary, studying one group (A or B) is equivalent to studying the other, since they are isomorphic and share all algebraic properties. However, defining algebraic properties can be subjective and may vary between groups. Some may even define algebraic properties as those that occur in all isomorphic groups.
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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.
 
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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?
 
  • #4
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.
 

What are isomorphic groups?

Isomorphic groups are groups that have the same structure and can be mapped onto each other in a way that preserves their group operations and elements.

How do you determine if two groups are isomorphic?

In order for two groups to be isomorphic, they must have the same number of elements, the same group operations, and the same structure. This can be determined by examining the group tables or by finding a bijective function between the two groups.

Why are isomorphic groups important in mathematics?

Isomorphic groups are important because they allow us to study one group and apply our findings to another group with the same structure. This saves time and effort in studying new groups and helps us understand the relationships between different groups.

Can isomorphic groups have different names and notations?

Yes, isomorphic groups can have different names and notations, as long as they have the same structure and can be mapped onto each other in a way that preserves their group operations and elements.

Can all groups be isomorphic to each other?

No, not all groups can be isomorphic to each other. Isomorphic groups must have the same structure and number of elements, so not all groups will meet these criteria.

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