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C-group

Class of mathematical groups

Note

the mathematical group theory concept

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
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 , 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 , 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 PSL2(q), PSU3(q), Sz(q) for q a power of 2.

References

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