Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

Lattice of Subgroups of a Group. Frattini Subgroup


Wojciech A. Trybulec
Warsaw University
Supported by RPBP.III-24.C1.

Summary.

We define the notion of a subgroup generated by a set of element of a group and two closely connected notions. Namely lattice of subgroups and Frattini subgroup. The operations in the lattice are the intersection of subgroups (introduced in [21]) and multiplication of subgroups which result is defined as a subgroup generated by a sum of carriers of the two subgroups. In order to define Frattini subgroup and to prove theorems concerning it we introduce notion of maximal subgroup and non-generating element of the group (see [9, page 30]). Frattini subgroup is defined as in [9] as an intersection of all maximal subgroups. We show that an element of the group belongs to Frattini subgroup of the group if and only if it is a non-generating element. We also prove theorems that should be proved in [1] but are not.

MML Identifier: GROUP_4

The terminology and notation used in this paper have been introduced in the following articles [13] [8] [22] [16] [2] [3] [14] [11] [23] [6] [7] [4] [19] [20] [5] [15] [10] [21] [17] [24] [18] [12] [1]

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. Sequences of ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] M. I. Kargapolow and J. I. Mierzlakow. \em Podstawy teorii grup. PWN, War\-sza\-wa, 1989.
[10] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[11] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[12] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[13] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[14] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.
[15] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[16] Wojciech A. Trybulec. Binary operations on finite sequences. Journal of Formalized Mathematics, 2, 1990.
[17] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Journal of Formalized Mathematics, 2, 1990.
[18] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[19] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Journal of Formalized Mathematics, 2, 1990.
[20] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[21] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Journal of Formalized Mathematics, 2, 1990.
[22] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[23] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[24] Stanislaw Zukowski. Introduction to lattice theory. Journal of Formalized Mathematics, 1, 1989.

Received August 22, 1990


[ Download a postscript version, MML identifier index, Mizar home page]