It is obvious that S(G)��MM. Moreover, S(G)��MS(G), since S(G) is cyclic. It follows that S(G)��MG. Therefore, G/(S(G)��M) sellekchem = S(G)/(S(G)��M)M/(S(G)��M). Let G-=G/(S(G)��M),S-(G)=S(G)/(S(G)��M), and M-=M/(S(G)��M). By N/C-theorem, NG-(S-(G))/CG-(S-(G))?Aut(S-(G)). That is, G-/CG-(S-(G))=S-(G)M-/CG-(S-(G)))?Aut(S-(G)). Note that Aut(S-(G)) is abelian since S-(G) is cyclic. Moreover, M-?S(G)M/S(G)=G/S(G) is a nonabelian simple group and S-(G)M-/CG-(S-(G))?(S-(G)M-/S-(G))/(CG-(S-(G))/S-(G)). Here S-(G)M-/S-(G)?M-. Therefore, one has CG-(S-(G))/S-(G)=1 or CG-(S-(G))/S-(G)=S-(G)M-/S-(G)=G-/S-(G). If CG-(S-(G))/S-(G)=1, it follows that S-(G)M-/S-(G)?Aut(S-(G)) is abelian, a contradiction. If CG-(S-(G))/S-(G)=G-/S-(G), then S-(G)��Z(G-). It follows that G-=S-(G)��M- and then MG; this contradicts that all maximal subgroups of G are nonnormal.

Thus, our assumption is not true, so ��(G) = S(G).It follows that G/��(G)PSL(2,5) or PSL(2,13).If ��(G) = 1, then GPSL(2,5) or PSL(2,13).Next, suppose that ��(G) �� 1. Let p be any prime divisor of |��(G)|. We claim that p2. Otherwise, assume that p > 2. Let T be a subgroup of ��(G) such that ��(G)/Tp. That is, ��(G/T)p. Then (G/T)/p(G/T)/��(G/T) = (G/T)/(��(G)/T)G/��(G)PSL(2,5) or PSL(2,13). Since p > 2 and Schur multipliers of both PSL(2,5) and PSL(2,13) are 2, we have that G/TPSL(2,5) �� p or PSL(2,13) �� p. Note that |(PSL(2,5) �� p)|>3 and |(PSL(2,13) �� p)|>3. It follows that |(G)|>3, a contradiction. Thus, p2, so ��(G) is a cyclic 2-group. If |��(G)| = 2n > 2, let L be a subgroup of ��(G) such that ��(G)/L2.

Then (G/L)/2(G/L)/��(G/L) = (G/L)/(��(G)/L)G/��(G)PSL(2,5) or PSL(2,13). We have that G/LSL(2,5) or SL(2,13). Let M be a subgroup of L such that L/M2. Then (G/M)/2(G/M)/(L/M)G/LSL(2,5) or SL(2,13). Since Schur multipliers of both SL(2,5) and SL(2,13) are trivial, we have that G/MSL(2,5) �� 2 or SL(2,13) �� 2; this contradicts that all maximal subgroups of G are nonnormal. Thus, |��(G)| = 2. It follows that GSL(2,5) or SL(2,13). Lemmas 10 and 11 combined together give Theorem 1. Acknowledgments The authors are grateful to the referees who gave valuable comments and suggestions. Jiangtao Shi was supported by NSFC (Grant nos. 11201401 and 11361075). Cui Zhang was supported by H.C. ?rsted Postdoctoral Fellowship at DTU (Technical University of Denmark).

In order to assess the influence of particulate matter (PM) on the Brefeldin_A air quality, ecosystems, human health, and climate changes, it is necessary to be aware of its chemical composition and size distribution [1�C3]. As humans are the most important recipients of environmental pollutants, differences in relationships between specific PM fractions (their concentrations) and morbidity and mortality of the human population must be taken into consideration.There is no concentration threshold for PM in the atmospheric air, below which the PM impact on the human health could be ignored [4].