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
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the projective special linear groups PSL2(9)
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the projective special linear groups PSL2(2 n) for n≥2
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the projective special linear groups PSL3(2 n) for n≥1
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the projective special unitary groups PSU3(2 n) for n≥2
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the Suzuki-Ree group Sz(2 2n+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 PSL
3(2
n) and PSU
3(2
n) 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 PSL
2(
q), PSU
3(
q), Sz(
q) for
q a power of 2.
