C-group

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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.

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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 finite non-abelian simple C-groups are

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 finite non-abelian simple ones consist of the finite non-abelian simple C-groups other than PSL3(2n) and PSU3(2n) for n≥2. 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), PSU3(q), Sz(q) for q a power of 2.

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