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On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

Let G be a finite group. The intersection graph Δ G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G , and two distinct vertices X and Y are adjacent if X Y 1 , where 1 denotes the trivial subgroup of order 1 . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection...

On the intersection graph of a finite group

Hossein Shahsavari, Behrooz Khosravi (2017)

Czechoslovak Mathematical Journal

For a finite group G , the intersection graph of G which is denoted by Γ ( G ) is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H K 1 . In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of Aut ( Γ ( G ) ) .

On totally inert simple groups

Martyn Dixon, Martin Evans, Antonio Tortora (2010)

Open Mathematics

A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.

On zero-symmetric nearrings with identity whose additive groups are simple

Wen-Fong Ke, Johannes H. Meyer, Günter F. Pilz, Gerhard Wendt (2024)

Czechoslovak Mathematical Journal

We investigate conditions on an infinite simple group in order to construct a zero-symmetric nearring with identity on it. Using the Higman-Neumann-Neumann extensions and Clay’s characterization, we obtain zero-symmetric nearrings with identity with the additive groups infinite simple groups. We also show that no zero-symmetric nearring with identity can have the symmetric group Sym ( ) as its additive group.

Simple group contain minimal simple groups.

Michael J. J. Barry, Michael B. Ward (1997)

Publicacions Matemàtiques

It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.

Simplicity of Neretin's group of spheromorphisms

Christophe Kapoudjian (1999)

Annales de l'institut Fourier

Denote by 𝒯 n , n 2 , the regular tree whose vertices have valence n + 1 , 𝒯 n its boundary. Yu. A. Neretin has proposed a group N n of transformations of 𝒯 n , thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that N n is generated by two groups: the group Aut ( 𝒯 n ) of tree automorphisms, and a Higman-Thompson group G n . We prove the simplicity of N n and of a family of its subgroups.

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