C-group

For the culture in ancient Nubia, see Nubian C-Group.

In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection.

The simple C-groups were determined by Suzuki (1965), and his classification is summarized by Gorenstein (1980, 16.4). The classification of C-groups was used in Thompson's classification of N-groups. The simple C-groups are

• the projective special linear groups PSL₂(p) for p a Fermat or Mersenne prime
• the projective special linear groups PSL₂(9)
• the projective special linear groups PSL₂(2ⁿ) for n≥2
• the projective special linear groups PSL₃(q) for q a prime power
• the Suzuki groups Sz(2²ⁿ⁺¹) for n≥1
• the projective unitary groups PU₃(q) for q a prime power

CIT-groups

The C-groups include as special cases the CIT-groups, that are groups in which the centralizer of any involution is a 2-group. These were classified by Suzuki (1961, 1962), and the simple ones consist of the C-groups other than PU₃(q) and PSL₃(q). The ones whose Sylow 2-subgroups are elementary abelian were classified in a paper of Burnside (1899), which was forgotten for many years until rediscovered by Feit in 1970.

TI-groups

The C-groups include as special cases the TI-groups (trivial intersection groups), that are groups in which any two Sylow 2-subgroups have trivial intersection. These were classified by Suzuki (1964), and the simple ones are of the form PSL₂(q), PU₃(q), Sz(q) for q a power of 2.