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C-group

{{Short description|Class of mathematical groups}}

{{About|the mathematical group theory concept|other uses|C group (disambiguation){{!}}C group}}

{{redirect|TI-group|the British company|TI Group|groups called TI|TI (disambiguation)}}

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 {{harvtxt|Suzuki|1965}}, and his classification is summarized by {{harvtxt|Gorenstein|1980|loc=16.4}}. 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 {{harvs|txt|last=Suzuki|year1=1961|year2=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 {{harvtxt|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 {{harvs|txt|last=Suzuki|year1=1964}}, and the simple ones are of the form PSL2(q), PSU3(q), Sz(q) for q a power of 2.

References

  • {{Citation |last=Burnside |first=William |author-link=William Burnside |chapter=On a class of groups of finite order |title=Proceedings of the Cambridge Philosophical Society |volume=18 |pages=269–276 |date=1899}}
  • {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite Groups | publisher=Chelsea | location=New York | isbn=978-0-8284-0301-6 | mr=569209 | year=1980}}
  • {{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki (mathematician) | title=Finite groups with nilpotent centralizers | doi=10.2307/1993556 | mr=0131459 | year=1961 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=99 | issue=3 | pages=425–470| jstor=1993556 | doi-access=free }}
  • {{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki (mathematician) | title=On a class of doubly transitive groups | jstor=1970423 | mr=0136646 | year=1962 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=75 | issue=1 | pages=105–145 | doi=10.2307/1970423| hdl=2027/mdp.39015095249804 | hdl-access=free }}
  • {{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki (mathematician) | title=Finite groups of even order in which Sylow 2-groups are independent | jstor=1970491 | mr=0162841 | year=1964 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=80 | issue=1 | pages=58–77 | doi=10.2307/1970491}}
  • {{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki (mathematician) | title=Finite groups in which the centralizer of any element of order 2 is 2-closed | jstor=1970569 | mr=0183773 | year=1965 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=82 | issue=1 | pages=191–212 | doi=10.2307/1970569}}

Category:Finite groups