c © TÜBİTAK Barely Transitive Groups

This is a survey article on barely transitive groups. It also involves some recent results in the case of a non-locally finite barely transitive group. Key Words: Locally finite group, barely transitive group, infinite permutation group. A group is said to satisfy the normalizer condition if every proper subgroup is distinct from its normalizer. It is well known that a finite group satisfies the normalizer condition if and only if it is nilpotent. Equivalently hypercentral. Kuros ̌ and Černikov asked in [19, Question XXI, page 327]; whether a group satisfying the normalizer condition need be hypercentral or not. The negative answer was first given by Heineken and Mohamed in [17]. They constructed groups G satisfying: (i) G is a countable locally finite p-group for a prime p. (ii) G/G ′ is isomorphic to Cp ∞ and G ′ is elementary abelian. (iii) Every proper subgroup of G is nilpotent and subnormal, so G satisfies the normalizer condition. (iv) Z(G) = {1} (v) The set of normal subgroups of G contained in G ′ is linearly ordered by set inclusion [15, page 334]. (vi) For every proper subgroup K of G, KG ′ is a proper subgroup of G [17, Lemm