The stabilizer group for an n-qubit state?|φ>is the set of all invertible local operators (ILO) g=g1?g2?…?gn, gi∈(G) L (2, C) such that|?φ>=g|?φ>. Recently, Gour et al. [Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58092204] presented that almost all n-qubit states?ψ>own a trivial stabilizer group when n≥5. In this article, we consider the case when the stabilizer group of an n-qubit symmetric pure state|?ψ>is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state?|φ>is nontrivial when n≤4. Then we present a class of n-qubit symmetric states|?φ>with a trivial stabilizer group when n≥5. Finally, we propose a conjecture and prove that an n-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5 under the conjecture we make, which confirms the main result of Gour et al. partly.