C-group

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title: "C-group" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["finite-groups"] description: "Class of mathematical groups" topic_path: "general/finite-groups" source: "https://en.wikipedia.org/wiki/C-group" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Class of mathematical groups ::

::callout[type=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

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::