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].
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]
-
Some examples of matrices
-
Deleting of rows and columns in a matrix
-
Sets of permutations
-
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
[
Download a postscript version,
MML identifier index,
Mizar home page]