C-group

For the culture in ancient Nubia, see 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 projective special linear groups PSL₂(p) for p a Fermat or Mersenne prime
the projective special linear groups PSL₂(9)
the projective special linear groups PSL₂(2ⁿ) for n≥2
the projective special linear groups PSL₃(q) for q a prime power
the Suzuki groups Sz(2²ⁿ⁺¹) for n≥1
the projective unitary groups PU₃(q) for q a prime power

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 simple ones consist of the C-groups other than PU₃(q) and PSL₃(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.

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), PU₃(q), Sz(q) for q a power of 2.

References

Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 81b:20002
Suzuki, Michio (1961), "Finite groups with nilpotent centralizers", Transactions of the American Mathematical Society 99: 425–470, doi:10.2307/1993556, ISSN 0002-9947, MR 0131459
Suzuki, Michio (1962), "On a class of doubly transitive groups", Annals of Mathematics. Second Series 75: 105–145, ISSN 0003-486X, JSTOR 1970423, MR 0136646
Suzuki, Michio (1964), "Finite groups of even order in which Sylow 2-groups are independent", Annals of Mathematics. Second Series 80: 58–77, ISSN 0003-486X, JSTOR 1970491, MR 0162841
Suzuki, Michio (1965), "Finite groups in which the centralizer of any element of order 2 is 2-closed", Annals of Mathematics. Second Series 82: 191–212, ISSN 0003-486X, JSTOR 1970569, MR 0183773