Finite Solvable Groups with Few Non-cyclic Subgroups

Abstract

Let G be a finite group and $$\delta (G)$$ denote the number of conjugate classes of allnon-cyclic subgroups of G. The symbol $$\pi (G)$$ denotes the set of the prime divisors of |G|. In Meng and Li (Sci Sin Math 44:939–944, 2014), it was proved that for a finite non-cyclic solvable group G, one always has $$\delta (G)\ge 2^{|\pi (G)|-2}$$ . The groups with $$\delta (G)\le |\pi (G)|+1$$ always are solvable and have been complete classified. Moreover, it was showed that a finite non-solvable group G with $$\delta (G)=|\pi (G)|+2$$ is isomorphic to $$A_5$$ or SL(2, 5). In this paper, we investigate the finite solvable groups with $$\delta (G)=|\pi (G)|+2$$ . For convenience, a group G is said to be a $$\delta \pi _2$$ -group if $$\delta (G)=|\pi (G)|+2$$ . In particular, we give a completely classification of the $$\delta \pi _2$$ -groups with $$|\pi (G)|=3,4$$ .