What is the meaning of the phrase "Up to isomorphism " in Abstract Algebra? What is the meaning of the phrase "Up to isomorphism " in Abstract Algebra in context of-


*

*GROUPS

*RINGS

*FIELDS

 A: In mathematics it matters less what things are than how they behave.
Things that behave in exactly the same way can be thought as being the same.
Isomorphisms make "behave in exactly the same way" a precise notion.
With the appropriate definition of isomorphism, this reasoning can be extended to other mathematical objects, such as vector spaces, topological spaces,  differentiable manifolds, etc.
A: The other answer and the comments have already said much.
Here is an example to think about. Consider the set $Z = \{0,1\}$. Consider the operation of addition modulo $2$. That is
$$\begin{align}
0 + 0 &= 0 \\
0 + 1 & = 1 \\
1 + 0 &= 1 \\
1 + 1 &= 0
\end{align}
$$
Under this operation $Z$ becomes a group. (You can easily check the axioms.) Now consider another group.
Let $Y = \{1, -1\}$. Then $Y$ and $X$ are different sets. Consider the operation of multiplication. That is
$$\begin{align}
1\cdot 1 &= 1 \\
1\cdot (-1) &= -1 \\
-1\cdot 1 &= -1 \\
(-1)\cdot(-1) &= 1
\end{align}
$$
Under this operation $Y$ is a group. It is technically a different group because $X$ and $Y$ are different sets. One has addition as operation and the other multiplication. But, as you can see from the above, the groups behave exactly the same way. There is no essential difference between the two groups. Saying that is kinda vague, so we have the concept of groups being isomorphic. Two isomorphic groups are exactly "essentially the same".
So here it the point: Take any two symbols/elements. Now make the set of these two elements into a group by defining some operation on the set. The fact is that no matter how you do this, you will always get a group that is isomorphic to one of the two above. That way, there is essentially only one group of order $2$ (i.e. a group with two elements). 
This is exactly what we mean when we say that there is, up to isomorphism, only one group of order $2$.
