It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.
@article{urn:eudml:doc:41316, title = {Simple group contain minimal simple groups.}, journal = {Publicacions Matem\`atiques}, volume = {41}, year = {1997}, pages = {411-415}, mrnumber = {MR1485492}, zbl = {0894.20019}, language = {en}, url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41316} }
Barry, Michael J. J.; Ward, Michael B. Simple group contain minimal simple groups.. Publicacions Matemàtiques, Tome 41 (1997) pp. 411-415. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41316/