The growth index $γ_L$ was proposed to investigate the possible deviation from the standard $Λ$CDM model and Einstein's gravity theory in a dynamical perspective. Recently, thanks to the measurement of the cosmic growth rate via the redshift-space distortion, one can understand the evolution of density contrast through $fσ_8(z)$, where $f(z)=d\ln δ/d \ln a$ is the growth rate of matter and $σ_8(z)$ is the rms amplitude of the density contrast $δ$ at the comoving $8h^{-1}$ Mpc scale. In this paper, we use the redshift space distortion data points to study the growth index on the bases of Einstein's gravity theory and a modified gravity theory under the assumption of $f=Ω_m(a)^{γ_L}$. The cosmic background evolution is fixed by the cosmic observations from the type Ia supernovae SNLS3, cosmic microwave background radiation data from {\it Planck} and baryon acoustic oscillations. Via the Markov Chain Monte Carlo method, we found the $γ_L$ values for Einstein's gravity with a cosmological constant, $w=constant$ dark energy and a modified gravity theory in the $1,2,3σ$ regions respectively: $0.675_{-0.0662-0.120-0.155}^{+0.0611+0.129+0.178}$, $0.745_{-0.0819-0.146-0.190}^{+0.0755+0.157+0.205}$ and $0.555_{-0.0167-0.0373-0.0516}^{+0.0193+0.0335+0.0436}$. In the Einstein's gravity theory, the values of growth index $γ_L$ show almost $2σ$ deviation from the theoretical prediction 6/11 for the $Λ$CDM model. However in the modified gravity framework, a deviation from the Einstein's relativity is not detected in $1σ$ region. That implies that the currently available cosmic observations don't expect an alternative modified gravity theory beyond the $Λ$CDM model under Einstein's gravity, but that the simple assumption of $f=Ω_m^{γ_L}$ should be improved.

to match the published version, to appear in PRD