Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

Transpose Matrices and Groups of Permutations


Katarzyna Jankowska
Warsaw University, Bialystok

Summary.

Some facts concerning matrices with dimension $2\times 2$ are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. Some proofs are based on [11] and [14].

MML Identifier: MATRIX_2

The terminology and notation used in this paper have been introduced in the following articles [15] [8] [20] [21] [5] [7] [6] [2] [18] [19] [4] [17] [13] [3] [1] [12] [10] [16] [9]

Contents (PDF format)

  1. Some examples of matrices
  2. Deleting of rows and columns in a matrix
  3. Sets of permutations
  4. Group of permutations

Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[10] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[11] Thomas W. Hungerford. \em Algebra, volume 73 of \em Graduate Texts in Mathematics. Springer-Verlag New York Inc., Seattle, Washington USA, Department of Mathematics University of Washington edition, 1974.
[12] Katarzyna Jankowska. Matrices. Abelian group of matrices. Journal of Formalized Mathematics, 3, 1991.
[13] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[14] Serge Lang. \em Algebra. PWN, Warszawa, 1984.
[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[16] Andrzej Trybulec and Agata Darmochwal. Boolean domains. Journal of Formalized Mathematics, 1, 1989.
[17] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[18] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[19] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Journal of Formalized Mathematics, 2, 1990.
[20] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received May 20, 1992


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