In this paper, we study the existence and asymptotic behavior of normalized ground states of the following Schrödinger system with critical Sobolev exponent: \(\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda _1u=|u|^4u+\mu vw& \text{ in }\ \mathbb {R}^3,\\ \displaystyle -\Delta v+\lambda _2v=|v|^4v+\mu wu& \text{ in }\ \mathbb {R}^3,\\ \displaystyle -\Delta w+\lambda _3w=|w|^4w+\mu uv& \text{ in }\ \mathbb {R}^3,\\ \int \limits _{\mathbb {R}^3}u^2=a^2,\ \ \int \limits _{\mathbb {R}^3}v^2=b^2,\ \ \int \limits _{\mathbb {R}^3}w^2=c^2,\\ \end{array}\right. \end{aligned}\) where \(a,b,c>0\) and \(\mu >0\) . We show that there exists a normalized ground state for \(0<\mu <\mu _0\) , the constant \(\mu _0\) will be explicitly given. Furthermore, we obtain the asymptotic behavior of the minimizers as \(\mu \rightarrow 0\) .
