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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