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)
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Functions and their basic properties.
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Functions from a set to a set.
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Some basic properties of sets.
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\em Podstawy teorii grup.
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Abelian groups, fields and vector spaces.
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Families of sets.
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Domains and their Cartesian products.
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Vectors in real linear space.
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Binary operations on finite sequences.
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Wojciech A. Trybulec.
Classes of conjugation. Normal subgroups.
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Groups.
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Wojciech A. Trybulec.
Non-contiguous substrings and one-to-one finite sequences.
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Pigeon hole principle.
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Subgroup and cosets of subgroups.
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Zinaida Trybulec.
Properties of subsets.
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Relations and their basic properties.
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1, 1989.
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Stanislaw Zukowski.
Introduction to lattice theory.
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1, 1989.
Received August 22, 1990
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