On the 2-exponent of a finite group

The main result of the author's 1968 PhD thesis is the following: Let T be a Sylow 2-subgroup of a finite simple group. If T has nilpotence class n, then Z(T) has exponent at most n2. This result was announced at a conference in 1970, but the proof was never published. We belatedly provide a proof here, along with proofs of several related results.