The group U4(3) is the image PSU4(3) of the special unitary group SU4(3) in PSL4(9). Thus this unitary group is a classical group (cf. Chapter 7 in [FGT] for a discussion of the classical groups and unitary groups in particular), but it exhibits sporadic behavior and is crucial in the study of various sporadic groups. For example, we saw in Exercise 9.4 that U4(3) is the stabilizer of a point in the rank 3 representation of the McLaughlin group. Further we find in Chapter 16 that U4(3) is a section in a 3-local of Suz.

In Section 45 we prove a uniqueness theorem for U4(3) characterizing the group in terms of the centralizer of an involution. In the end we construct a geometric complex for the group which we prove to be isomorphic to the complex of singular points and lines in the 4-dimensional unitary space over the field of order 9. This last part of the proof can also be used in Exercise 9.4 to identify the point stabilizer of Mc in terms of the information in Lemma 24.7.

In Chapter 16 we use the characterization of Section 45 to pin down the structure of the centralizer of a certain subgroup of order 3 in groups of type Suz. This result is used in turn in the proof of the uniqueness of Suz and in 26.6 to determine the structure of the centralizer of the corresponding subgroup of order 3 in C01, while 26.6 is used in 32.4 to establish the existence of a subgroup of the Monster of type F24.