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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?
quasar987 said:Provided you define "algebraic properties" correctly, then yes.
Yes: usually an algebraic property is defined as a property which is preserved under isomorphism.tgt said:Would some even define algebraic properties to be those that occur in all 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.
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.
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.
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.
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.