# Group of $2\times2$ matrices that are isomorphic to $SU(2)$

Studying physics I encountered the $$SU(2)$$ group, in the context in which I use it $$SU(2)$$ is the group of the $$2\times2$$ unitary matrices with determinant equal to one. Out of curiosity, I searched the definition of group and I discovered that every other group that has an isomorphism with $$SU(2)$$ can be mathematically considered the same group. This definition creates some doubts in my understanding of physics, and in particular it's important for me to know if there are other sets of $$2\times2$$ complex-numbers matrices that are isomorphic to $$SU(2)$$ or if the set of unitary matrices with determinant equal to one is the only one.

Since I'm not a mathematician it's possible that the question is not clear, if that is the case fell free to ask clarifications.

• $SU(3)$ is a group. It has a lot of subgroups isomorphic to $SU(2)$, but these subgroups aren't all the same. May 6 '20 at 8:17
• Is the "set of unitary matrices with determinant equal to one" a "set of $2\times 2$ complex-numbers matrices that are isomorphic to $\Bbb C^2$ ?" May 6 '20 at 8:18
• I'm really sorry that was an error in the question, I edited so now it's clear May 6 '20 at 8:21

Yes, there are others. Let $$A$$ be a $$2\times2$$ invertible matrix with complex entries. Then$$\{AMA^{-1}\mid M\in SU(2)\}$$is a group of $$2\times2$$ invertible matrices which is isomorphic to $$SU(2)$$ but which, in general, is distinct from it.