# 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 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 SuzukiĀ (1961, 1962), and the simple ones consist of the C-groups other than PU_{3}(*q*) and PSL_{3}(*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.

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 SuzukiĀ (1964), and the simple ones are of the form PSL_{2}(*q*), PU_{3}(*q*), Sz(*q*) for *q* a power of 2.