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The only 4-transitive groups are the symmetric groups S k for k at least 4, the alternating groups A k for k at least 6, and the Mathieu groups M 24, M 23, M 12, and M 11. ( Cameron 1999 , p. 110) The full proof requires the classification of finite simple groups , but some special cases have been known for much longer.
M 11 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. M 11 is a sharply 4-transitive permutation group on 11 objects.
Some Christian bodies are large (e.g. Catholics, Orthodox, Pentecostals and nondenominationals, Anglicans or Baptists), while others are just a few small churches, and in most cases the relative size is not evident in this list except for the denominational group or movement as a whole (e.g. Church of the East, Oriental Orthodox Churches, or ...
Choir dress of a cardinal, in scarlet Cardinals are senior members of the clergy of the Catholic Church. As titular members of the clergy of the Diocese of Rome, they serve as advisors to the pope, who is the bishop of Rome. They are typically ordained bishops and generally hold important roles within the church, such as leading prominent archdioceses or heading dicasteries within the Roman ...
This is a list of the largest Protestant denominations. It aims to include sizable Protestant communions, federations, alliances, councils, fellowships, and other denominational organisations in the world and provides information regarding the membership thereof.
The list includes the Catholic Church (including Eastern Catholic Churches), Protestant denominations with at least 0.2 million members, the Eastern Orthodox Church (and its offshoots), Oriental Orthodox Churches (and their offshoots), Nontrinitarian Restorationism, independent Catholic denominations, Nestorianism and all the other Christian ...
This list may not reflect recent changes. A. Association of Independent Methodists; B. Bible Christian Faith Church; ... Free Presbyterian Church of North America;
M 12 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873).It is a sharply 5-transitive permutation group on 12 objects. Burgoyne & Fong (1968) showed that the Schur multiplier of M 12 has order 2 (correcting a mistake in (Burgoyne & Fong 1966) where they incorrectly claimed it has order 1).