We deal with the stability problem of the scalar linear time invariant (LTI) stochastic system driven by fractional Brownian motion (fBm). Firstly, the necessary and sufficient conditions are provided for the almost sure asymptotic stability and pth moment asymptotic stability by means of the largest Lyapunov exponent and the Lyapunov exponent of the pth mean, respectively. Furthermore, we obtain the large deviations results for this fractional process. It has been shown that the Hurst parameter affects the stability conclusions and the large deviations. Interestingly, the large deviations always happen for the considered system when 1/2