Lattice of Subgroups of a Group

We define the notion of a subgroup generated by a set of elements of a group and two closely connected notions, namely lattice of subgroups and the Frattini subgroup. The operations on the lattice are the intersection of subgroups (introduced in [18]) 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 the Frattini subgroup and to prove theorems concerning it we introduce notion of maximal subgroup and non-generating element of the group (see page 30 in [6]). The Frattini subgroup is defined as in [6] as an intersection of all maximal subgroups. We show that an element of the group belongs to the 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.