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A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M 11, M 12, M 22, M 23 and M 24 introduced by 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.
See Rubik's Cube group. In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout ...
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics ...
The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1]
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G.
Here the order of the group is n! while the n × n matrices form a vector space of dimension n 2. As soon as n is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16); this relation means that the module for the group algebra is not faithful.
M 12 has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M 11. Identifying the 12 points with the projective line over the field of 11 elements, M 12 is generated by the permutations of PSL 2 (11) together with the permutation (2,10)(3,4)(5,9)(6,7).
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