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Permutations which make transitive groups primitive

100%

Lopes P.

Open Mathematics

2009

tom 7

nr 4

650-659

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.

Equivalence classes of colorings

51%

Ge J., Jablan S., Kauffman L., Lopes P.

Banach Center Publications

2014

tom 103

nr 1

63-76

For any link and for any modulus m we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).

Ograniczanie wyników

1 Banach Center Publications

1 Open Mathematics

2 Lopes P.

1 Ge J.

1 Jablan S.

1 Kauffman L.

1 2014

1 2009

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Permutations which make transitive groups primitive

Equivalence classes of colorings