Permuting transfer-closed conjugacy-closed-to-retract

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup H of a group G is termed permuting transfer-closed normal-to-complemented in G if the following is true.

Suppose K_1, K_2, \dots K_n and H = H_1, H_2, \dots H_n, H_{n+1} are subgroups such that:

Then, if H_{n+1} is a conjugacy-closed subgroup of K_n, then H_{n+1} is a retract of K_n: it has a normal complement in K_n.

Relation with other properties

Stronger properties

Weaker properties