Simple groups contain minimal simple groups
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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.
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Barry, M. J. J.; Ward, M. B. «Simple groups contain minimal simple groups». Publicacions Matemàtiques, 1997, vol.VOL 41, núm. 2, p. 411-5, http://raco.cat/index.php/PublicacionsMatematiques/article/view/37903.