C-group

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 Suzuki groups Sz(22n+1) for n≥1