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 , and his classification is summarized by . The classification of C-groups was used in Thompson's classification of N-groups. The finite non-abelian simple C-groups are

• the projective special linear groups PSL₂(p) for p a Fermat or Mersenne prime, and p≥5
• the projective special linear groups PSL₂(9)
• the projective special linear groups PSL₂(2ⁿ) for n≥2
• the projective special linear groups PSL₃(2ⁿ) for n≥1
• the projective special unitary groups PSU₃(2ⁿ) for n≥2
• the Suzuki groups Sz(2²ⁿ⁺¹) for n≥1

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 , 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 , 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 , and the simple ones are of the form PSL₂(q), PSU₃(q), Sz(q) for q a power of 2.

References