C-group

Short description: Class of mathematical groups

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). 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 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, 18, pp. 269–276 
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6 
• Suzuki, Michio (1961), "Finite groups with nilpotent centralizers", Transactions of the American Mathematical Society 99 (3): 425–470, doi:10.2307/1993556, ISSN 0002-9947 
• Suzuki, Michio (1962), "On a class of doubly transitive groups", Annals of Mathematics, Second Series 75 (1): 105–145, doi:10.2307/1970423, ISSN 0003-486X 
• Suzuki, Michio (1964), "Finite groups of even order in which Sylow 2-groups are independent", Annals of Mathematics, Second Series 80 (1): 58–77, doi:10.2307/1970491, ISSN 0003-486X 
• Suzuki, Michio (1965), "Finite groups in which the centralizer of any element of order 2 is 2-closed", Annals of Mathematics, Second Series 82 (1): 191–212, doi:10.2307/1970569, ISSN 0003-486X