This article is about the mathematical group theory concept. For other uses, see C group.
"TI-group" redirects here. For the British company, see TI Group. For groups called TI, see TI (disambiguation).
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 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 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.
References
Burnside, William (1899), "On a class of groups of finite order", Proceedings of the Cambridge Philosophical Society, vol. 18, pp. 269–276