Isomorphic groups that have different properties?

[tgt]

What are some properties apart from the actual names of the elements that differ between isomorphic groups?

 

[radou]

What are some properties apart from the actual names of the elements that differ between isomorphic groups?

Your question actually is - what, in general, are the differences between different groups?

Elements and the defined binary operation.

 

[HallsofIvy]

that's pretty much it. Two isomorphic groups (or fields or... pretty much anything isomorphic) differ only in the "labeling": what you name the elements and operations.

 

[mathwonk]

in practice it is sometimes a little more subtle than it seems.

e.g. we usually give a cyclic group in the form (Z/n,+) but may not notice that this gives not only the group but also a distinguished generator, namely 1.

the group( (Z/p)*, .) of units in the ring Z/p is also cyclic when p is a prime, but it may not be immediately clear what a generator is. moreover since there are many generators, (Z/p)* and no one is naturally distinguished, there is no completely natural way to choose an isomorphism between the groups (Z/p)* and Z/(p-1).

i.e. in a sense, ({1,2,3,...,p-1}, . ) are just different names for ({0,1,2,...,p-2},+) but it is not clear which new nAME CORRESPONDS TO WHICH OLD NAME.

e.g. when p = 7, (Z/7)* is isomorphic to Z/6, and the least generator is 3, perhaps it is natural to associate n with 3^n, for n=0,...,5, but I am not sure this generator is always best.