group action is sharply 2-transitive if such g ∈ G {\displaystyle g\in G} is unique. A 2-transitive group is a group such that there exists a group action...

and more generally multiply transitive groups is well-studied in finite group theory. An action is sharply n-transitive when the action on tuples without...

Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered. Sometimes...

characteristic is not 2). The action of the projective linear group on the projective line is sharply 3-transitive (faithful and 3-transitive), so the map is one-to-one...

1882, §39–40). A transitive subgroup of Sn is a subgroup whose action on {1, 2, ,..., n} is transitive. For example, the Galois group of a (finite) Galois...

and d2 = 24. The action of a group G on a set M is said to be transitive if, for every two elements s, t of M, there is some group element g such that...

as group theory, the Higman–Sims group HS is a sporadic simple group of order    44,352,000 = 29 · 32 · 53 · 7 · 11 ≈ 4×107. The Schur multiplier has...

26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer...

discovered. The Schur multiplier and the outer automorphism group are both trivial. M11 is a sharply 4-transitive permutation group on 11 objects. It admits...

Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists...

generates a one-parameter subgroup. If a group G acts on a space V, then a surface S ⊂ V is a surface of transitivity if S is invariant under G (i.e., ∀g ∈...

sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is...

power order. The classification of 2-transitive permutation groups. The classification of rank 3 permutation groups. The Sims conjecture Frobenius's conjecture...

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 &...

22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a permutation group on 24 points. All the...

ISBN 0-8186-0591-X. L. Babai, Local expansion of vertex-transitive graphs and random generation in finite groups, Proc. 23rd STOC (1991), 164–174. Frank Celler;...

alternating group A 4 {\displaystyle A_{4}} (and also the symmetric group S 4 {\displaystyle S_{4}} ) on four letters. It is also a transitive subgroup of...

orthogonal group O(n + 1) acts transitively on the unit sphere Sn, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the...

permutes the 24 pairs of the frame. Co0 can be shown to be transitive on Λ4. Conway multiplied the order 212|M24| of N by the number of frames, the latter...

characterizes elementary abelian groups among all finite groups: if G is a finite group with identity e such that Aut(G) acts transitively on G \ {e}, then G is...

subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication...

In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some...

sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer...

exactly the vertex-transitive graphs whose symmetry group includes a transitive cyclic group. The endomorphism ring of the abelian group Z/nZ is isomorphic...

containing a given maximal torus, and they are permuted simply transitively by the Weyl group (acting by conjugation). A choice of Borel subgroup determines...

diffeomorphism group acts transitively on the configuration space F k M {\displaystyle F_{k}M} and the action on M {\displaystyle M} is multiply transitive (Banyaga...

chambers is this: Theorem: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl...

simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere S 5 {\displaystyle...

{\displaystyle \mathrm {SU} (n),} with n ≥ 2. {\displaystyle n\geq 2.} This group acts transitively on the unit sphere S 2 n − 1 {\displaystyle S^{2n-1}} inside C...

{\displaystyle j=1,2,3} ⁠. (In other words: the action of the Möbius group on the Riemann sphere is sharply 3-transitive.) There are several ways to determine f...