Another existence and uniqueness proof of the tits group

In this article we give a self-contained existence and Uniqueness proof for the Tits simple group T. Parrott gave the first uniqueness proof. Whereas Tits' and Parrott's results employ the theory of finite groups of Lie type, our existence and uniqueness proof follows from the general algorithms and uniqueness criteria for abstract finite simple groups described in the first author's book [11]. All we need from the previous papers is the fact that the centralizer H of the Tits group T is an extension of a 2-group J with order 2(9) and nilpotency class 3 by a Frobenius group F of order 20 such that the center Z(H) has order 2 and any Sylow 5-subgroup. Q of H has a centralizer C-J(Q)<= Z (H).