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The earlier papers in this series have reduced the problem of characterizing the minimal simple groups to several stubborn special cases. This paper handles some of these special cases. Almost all the difficulties of this paper involve groups of order 2 α .3 δ for which min(α,δ) is rather small. This paper is a continuation of its predecessors.doi:10.2140/pjm.1974.50.215 fatcat:osvficj6gfhbpctcrlhmymgsjq