A trio is a set of 3 disjoint octads of the Golay code. The subgroup fixing a trio is the trio group 26:(PSL(2,7) x S3), order 64512, transitive and imprimitive.
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An octern is a certain partition of the 24 points into 8 blocks of 3. The subgroup fixing an octern is the octern group isomorphic to PSL2(7), of order 168, simple, transitive and imprimitive. It was the last maximal subgroup of M24 to be found.
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A duum is a pair of complementary dodecads (12 point sets) in the Golay code. The subgroup fixing a duad is M12:2, order 190080, transitive and imprimitive. This subgroup was discovered by Frobenius. The subgroup M12 acts differently on 2 sets of 12, reflecting the outer automorphism of M12.
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A permutation group G acting transitively on a non-empty finite set M is imprimitive if there is some nontrivial set partition of M that is preserved by the action of G, where "nontrivial" means that the partition isn't the partition into a singleton sets nor the partition with only one part. Otherwise, if G is transitive but does not preserve any nontrivial partition of M, the group G is primitive. For example, the group of symmetries of a square is primitive on the vertices: if they are numbered 1, 2, 3, 4 in cyclic order, then the partition {{1, 3}, {2, 4}} into opposite pairs is preserved by every group element. On the other hand, the full symmetric group on a set M is always imprimitive.
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Assign a zero when there is no influence. Thus obtain the weighted column stochastic supermatrix. #Compute the limit priorities of the stochastic supermatrix according to whether it is irreducible (primitive or imprimitive [cyclic]) or it is reducible with one being a simple or a multiple root and whether the system is cyclic or not. Two kinds of outcomes are possible.
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Some notable and highly-cited examples of this work are as follows. Liebeck, Saxl and Praeger gave a relatively simple and self-contained proof of the O'Nan–Scott theorem. It had long been known that every maximal subgroup of a symmetric group or alternating group was intransitive, imprimitive, or primitive, and the same authors in 1988 gave a partial description of which primitive subgroups could occur.
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The theory was vastly extended and refined by Gauss in Section V of Disquisitiones Arithmeticae. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. He replaced Lagrange's equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a group under the composition operation. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares.
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Bareis then completed part of her graduate work at Bryn Mawr College during 1897–1899, and later at Columbia University in that same year She took a brief break from school to work as a teacher at Miss Roney's School in Philadelphia where she would continue to work for the next six years. Two years after leaving Columbia University, she returned to Bryn Mawr College to continue her work on a mathematics degree. In 1906, Bareis enrolled in the graduate program at Ohio State University and after three years received the first Ph.D in mathematics from the university. Her dissertation, "Imprimitive Substitution Groups of Degree Sixteen", was directed by Harry W. Kuhn.
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En route to this goal he introduced the notion of the order of an element of a group, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive and proved some important theorems relating these concepts, such as However, he got by without formalizing the concept of a group, or even of a permutation group. The next step was taken by Évariste Galois in 1832, although his work remained unpublished until 1846, when he considered for the first time what is now called the closure property of a group of permutations, which he expressed as The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy and Camille Jordan, both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Among other things, Jordan defined a notion of isomorphism, still in the context of permutation groups and, incidentally, it was he who put the term group in wide use. The abstract notion of a group appeared for the first time in Arthur Cayley's papers in 1854.
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