Isomorphic groups that have different properties?

1. Aug 22, 2008

tgt

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

2. Aug 22, 2008

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

Elements and the defined binary operation.

3. Aug 23, 2008

HallsofIvy

Staff Emeritus
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.

4. Aug 23, 2008

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.