Solitary Subgroups

Abstract

We call a subgroup A of a finite group G a solitary subgroup of G if G does not contain another isomorphic copy of A. We call a normal subgroup A of a finite group G a normal solitary subgroup of G if G does not contain another normal isomorphic copy of A. The property of being (normal) solitary can be viewed as a strengthening of the requirement that A is normal in G. We derive various results on the existence of (normal) solitary subgroups.

Finite groups with all minimal subgroups solitary

“…A subgroup H of a group G is called solitary when for every subgroup K of G such that H ∼ = K, we have that H = K. This concept was introduced by Thévenaz in [12] under the name of strongly characteristic subgroup and studied by Kaplan and Levy in [10]. It is clear that solitary subgroups are characteristic and thus normal.…”

Subgrupos solitarios de grupos finitos

“…We say that a subgroup H of a group G is solitary when no other subgroup of G is isomorphic to H. A normal subgroup H of a group G is said to be normal solitary when no other normal subgroup of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N. Solitary subgroups, normal solitary subgroups, and quotient solitary subgroups have been recently studied by authors like Thévenaz [Thé93], who named the solitary subgroups as strongly characteristic subgroups, Kaplan and Levy [KL09,Lev14], Tȃrnȃuceanu [Tȃr12b,Tȃr12a], and Atanasov and Foguel [AF12].…”

“…In this chapter we deepen into the study of the lattices of solitary subgroups and quotient solitary subgroups developed by Kaplan and Levy [KL09] and by Tȃrnȃuceanu [Tȃr12b] and we check Summary that, even though these lattices consist of normal subgroups, they are not sublattices of the lattice of normal subgroups. We also check that the set of all normal solitary subgroups does not constitute a lattice, which motivates the introduction of the concept of subnormal solitary subgroup as a more suitable tool to deal with lattice properties.…”

“…Direm que un subgrupo H d'un grup G és solitari quan cap altre subgrup de G no és isomorf a H. Un subgrup normal H d'un grup G es diu normal solitari quan cap altre subgrup normal de G no és isomorf a H. Un subgrup normal N d'un grup G es diu que és solitari per a quocients quan cap altre subgrup normal K de G no dóna un quocient isomorf a G/N. Els subgrups solitaris, els subgrups solitaris normals i els subgrups solitaris per a quocients han sigut recentment estudiats per autors com Thévenaz [Thé93], qui batejà els subgrups solitaris com a subgrups fortament característics, Kaplan i Levy [KL09,Lev14], Tȃrnȃuceanu [Tȃr12b,Tȃr12a] i Atanasov i Foguel [AF12].…”

“…En este capítulo se profundiza en el estudio de los retículos de subgrupos solitarios y solitarios para cocientes lle-Resumen vado a cabo por Kaplan y Levy [KL09] y por Tȃrnȃuceanu [Tȃr12b] y se comprueba que, a pesar de que estos retículos constan de subgrupos normales, no son subretículos del retículo de subgrupos normales. También comprobamos que el conjunto de subgrupos normales solitarios no constituye un retículo, lo que motiva la introducción del concepto de subgrupo subnormal solitario como herramienta más adecuada para tratar propiedades reticulares.…”