A note on TI-subgroups (Q1117034)

Let A be an abelian 2-subgroup of a finite group G such that A is a TI- set. Then the authors show that one of the following holds: (1) \(| \Omega_ 2(A)/\Omega_ 1(A)| \leq 2\), (ii) \(A\cong {\mathbb{Z}}_ 4\times {\mathbb{Z}}_ 4\), (iii) \(\exp (A)=4\), \(<\Omega_ 1(A)^ G>\) is abelian and a description of \(<A^ G>\) can be given, or (iv) \(<A^ G>\) is abelian. For all types examples really occur. The authors claim that a TI-group of type (ii) can only occur in \(L_ 3(4)\).