Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

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

- 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.

Contents (PDF format)

- [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.

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