Erratum to: “Solitary quotients of finite groups”
In this short note a correct proof of Theorem 3.3 from [Tărnăuceanu M., Solitary quotients of finite groups, Cent. Eur. J. Math., 2012, 10(2), 740–747] is given.
Solitary quotients of finite groups
We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.
Breaking points in the poset of conjugacy classes of subgroups of a finite group
We determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion -groups. A generalization of this property is also studied.
Isolated subgroups of finite abelian groups
We say that a subgroup is isolated in a group if for every we have either or . We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.
On a group-theoretical generalization of the Gauss formula
We discuss a group-theoretical generalization of the well-known Gauss formula involving the function that counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
On the total number of principal series of a finite abelian group.
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