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A ^-group is a group in which normality is transitive and a minimal non-^-group is a group which is not a ^-group but all of whose proper subgroups are ^-groups. In this note all the finite minimal non-St-groups are determined and it is shown that this classification also accounts for all the minimal non2-groups which are either locally finite or 2-groups. 1* S-groups and non-2^groups* Let X denote the class of all groups in which normality is a transitive relation: in other words a group G belongs to X if and only if H groups. The problem of determining all the finite minimal non-X-groups has been studied by several authors when 36 is the class of abelian, nilpotent, supersoluble or soluble groups. Our principal object here is the classification of all the finite minimal non-S-groups or, what is obviously the same thing, the finite minimal non-S-groups. Although the nature and even the existence of infinite minimal non-ϊ-groups is problematic, our classification does account for all minimal non-2groups which are locally finite or 2-groups or Si-groups (in the sense of Euros [5J, vol. 2, § 57). The finite minimal non-X-groups. Such groups are all soluble and it turns out that they fall naturally into seven types, which will now be listed under four headings, beginning with the primary groups.

Groups which are minimal with respect to normality being intransitive