I came across this post on Math Stack Exchange where OP asks about two groups which have same order but are not isomorphic.
And example given is the Klein-4 group, and $\mathbb{Z}_{4}$ among many others. But no one explained why these are not isomorphic. For something to be isomorphic, we NEED to define a function between one group, and the other right? How can we just look at groups and say one is isomorphic to the other?
Is there a hole in my intuitive understanding for what it means for two groups to be isomorphic to one another?
Best Answer
In practice, it is much easier to determine that two groups are not isomorphic than to show that they are. To show that they're not isomorphic, it suffices to find a single different property.
For example, there is no element of the Klein $4$ group that has order $4$. The largest order of an element is $2$. The cyclic group $\mathbb{Z}/4\mathbb{Z}$ has an element of order $4$. Thus these two groups are not isomorphic.
Other useful aspects that can frequently be used to show that two groups are not isomorphic include cardinality, conjugacy classes, sizes and numbers of subgroups, among others.
But it is still possible to find pairs of groups that share practically all easily computable aspects noted above, but still aren't isomorphic. This is a pretty annoying problem and very slow to compute for general finite groups. For general infinite groups, the isomorphism problem is undecidable! (M.O. Rabin, Recursive unsolvability of group theoretic problems. Ann. Math 67:172--194, 1958).