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C group

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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.

Contents

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

    C-group Wikipedia
    (Text) CC BY-SA