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Papers by Mohammad Naghshinehfard

Rocky Mountain Journal of Mathematics, 2018

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap ... more We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases). 1. Introduction. Let G be a group. For each x ∈ G, the cyclizer of x is defined as Cyc G (x) = {y ∈ G | ⟨x, y⟩ is cyclic}. In addition, the cyclizer of G is defined by Cyc(G) = ∩ x∈G Cyc G (x). Cyclizers were introduced by Patrick and Wepsic in [15] and studied in [1, 2, 3, 9, 14, 15]. It is known that Cyc(G) is always cyclic and that Cyc(G) ⊆ Z(G). In particular, Cyc(G) G. The cyclic graph (respectively, weak cyclic graph) of a group G is the simple graph with vertex-set G\Cyc(G) (respectively, G\{1}) such that two distinct vertices x and y are adjacent if and only if ⟨x, y⟩ is cyclic. The cyclic graph and weak cyclic graph of G are denoted by Γ c (G) and Γ w c (G), respectively. From the explanation above, Γ c (G) (respectively, Γ w c (G)) is the null graph if G is cyclic (respectively, trivial). Thus, we will assume that G is non-cyclic (respectively, nontrivial) when working with Γ c (G) (respectively, Γ w c (G)). A graph is planar if it can be drawn in the plane in such a way that its edges intersect only at the end vertices. Recall that a subdivision of an edge {u, v} in a graph Γ is the replacement of the edge {u, v} in Γ with two new edges {u, w} and {w, v} in which w is a new vertex.

Bulletin of the Korean Mathematical Society, 2011

In the present paper we introduce the lower autocentral series of autocommutator subgroups of a g... more In the present paper we introduce the lower autocentral series of autocommutator subgroups of a given group. Following our previous work on the subject in 2009, it is shown that every finite abelian group is isomorphic with n th-term of the lower autocentral series of some finite abelian group.

Journal of Algebra and Its Applications, 2017

We compute the autocentral and autoderived series of a finite abelian group [Formula: see text] c... more We compute the autocentral and autoderived series of a finite abelian group [Formula: see text] corresponding to several groups of automorphisms [Formula: see text]. As a result, we determine the autonilpotency and autosolvability of [Formula: see text] with respect to [Formula: see text].

Journal of Mathematical …, 2009

Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α) −1 (g... more Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α) −1 (g −1 g α) β is the autocommutator of the element g ∈ G and α, β ∈ Aut(G) of weight 3. Also, we define K 2 (G) =< [g, α, β] : g ∈ G, α, β ∈ Aut(G) > to be the third term of the lower autocentral series of subgroups of G. In this paper, it is shown that every finite abelian group is isomorphic to the third term of the autocentral series of some finite abelian group.

International Journal of Pure and Apllied Mathematics, 2013

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the comm... more For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].

All finite groups with toroidal or projective cyclic graphs are classified. Indeed, it is shown t... more All finite groups with toroidal or projective cyclic graphs are classified. Indeed, it is shown that the only finite groups with projective cyclic graphs are S 3 × Z 2 , D 14 , QD 16 and which all have toroidal cyclic graph too. Also, D 16 is characterized as the only finite group whose cyclic graph is toroidal but not projective

A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer auto... more A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer automorphism group is given. Also the classification of those finite groups G such that K(G) ∼ = H if H is a centerless finite group with cyclic outer automorphism group and possible solutions G if |Z(H)| = 2 and H has a cyclic outer automorphism group is presented.

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the comm... more For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].

Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α ) −1 (... more Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α ) −1 (g −1 g α ) β is the autocommutator of the element g ∈ G and α, β ∈ Aut(G) of weight 3. Also, we define K 2 (G) =< [g, α, β] : g ∈ G, α, β ∈ Aut(G) > to be the third term of the lower autocentral series of subgroups of G. In this paper, it is shown that every finite abelian group is isomorphic to the third term of the autocentral series of some finite abelian group.

Note Mat. 31(2) (2011), 9 - 16

A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer auto... more A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer automorphism group is given. Also the classification of those finite groups GGG such that K(G)congHK(G)\cong HK(G)congH if HHH is a centerless finite group with cyclic outer automorphism group and possible solutions GGG if ∣Z(H)∣=2|Z(H)|=2∣Z(H)∣=2 and HHH has a cyclic outer automorphism group is presented.

In the present paper we introduce the lower autocentral series of autocommutator subgroups of a g... more In the present paper we introduce the lower autocentral series of autocommutator subgroups of a given group. Following our previous work on the subject in 2009, it is shown that every finite abelian group is isomorphic with n th -term of the lower autocentral series of some finite abelian group.

Drafts by Mohammad Naghshinehfard

We compute the autocentral and autoderived series of a fi nite abelian group G corresponding to s... more We compute the autocentral and autoderived series of a fi nite abelian group G corresponding to several groups of automorphisms A. As a result, we determine the autonilpotency and autosolvability of G with respect to A.

Rocky Mountain Journal of Mathematics, 2018

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap ... more We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases). 1. Introduction. Let G be a group. For each x ∈ G, the cyclizer of x is defined as Cyc G (x) = {y ∈ G | ⟨x, y⟩ is cyclic}. In addition, the cyclizer of G is defined by Cyc(G) = ∩ x∈G Cyc G (x). Cyclizers were introduced by Patrick and Wepsic in [15] and studied in [1, 2, 3, 9, 14, 15]. It is known that Cyc(G) is always cyclic and that Cyc(G) ⊆ Z(G). In particular, Cyc(G) G. The cyclic graph (respectively, weak cyclic graph) of a group G is the simple graph with vertex-set G\Cyc(G) (respectively, G\{1}) such that two distinct vertices x and y are adjacent if and only if ⟨x, y⟩ is cyclic. The cyclic graph and weak cyclic graph of G are denoted by Γ c (G) and Γ w c (G), respectively. From the explanation above, Γ c (G) (respectively, Γ w c (G)) is the null graph if G is cyclic (respectively, trivial). Thus, we will assume that G is non-cyclic (respectively, nontrivial) when working with Γ c (G) (respectively, Γ w c (G)). A graph is planar if it can be drawn in the plane in such a way that its edges intersect only at the end vertices. Recall that a subdivision of an edge {u, v} in a graph Γ is the replacement of the edge {u, v} in Γ with two new edges {u, w} and {w, v} in which w is a new vertex.

Bulletin of the Korean Mathematical Society, 2011

In the present paper we introduce the lower autocentral series of autocommutator subgroups of a g... more In the present paper we introduce the lower autocentral series of autocommutator subgroups of a given group. Following our previous work on the subject in 2009, it is shown that every finite abelian group is isomorphic with n th-term of the lower autocentral series of some finite abelian group.

Journal of Algebra and Its Applications, 2017

We compute the autocentral and autoderived series of a finite abelian group [Formula: see text] c... more We compute the autocentral and autoderived series of a finite abelian group [Formula: see text] corresponding to several groups of automorphisms [Formula: see text]. As a result, we determine the autonilpotency and autosolvability of [Formula: see text] with respect to [Formula: see text].

Journal of Mathematical …, 2009

Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α) −1 (g... more Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α) −1 (g −1 g α) β is the autocommutator of the element g ∈ G and α, β ∈ Aut(G) of weight 3. Also, we define K 2 (G) =< [g, α, β] : g ∈ G, α, β ∈ Aut(G) > to be the third term of the lower autocentral series of subgroups of G. In this paper, it is shown that every finite abelian group is isomorphic to the third term of the autocentral series of some finite abelian group.

International Journal of Pure and Apllied Mathematics, 2013

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the comm... more For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].

All finite groups with toroidal or projective cyclic graphs are classified. Indeed, it is shown t... more All finite groups with toroidal or projective cyclic graphs are classified. Indeed, it is shown that the only finite groups with projective cyclic graphs are S 3 × Z 2 , D 14 , QD 16 and which all have toroidal cyclic graph too. Also, D 16 is characterized as the only finite group whose cyclic graph is toroidal but not projective

A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer auto... more A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer automorphism group is given. Also the classification of those finite groups G such that K(G) ∼ = H if H is a centerless finite group with cyclic outer automorphism group and possible solutions G if |Z(H)| = 2 and H has a cyclic outer automorphism group is presented.

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the comm... more For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].

Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α ) −1 (... more Let G be a group and Aut(G) be the group of automorphisms of G. Then [g, α, β] = (g −1 g α ) −1 (g −1 g α ) β is the autocommutator of the element g ∈ G and α, β ∈ Aut(G) of weight 3. Also, we define K 2 (G) =< [g, α, β] : g ∈ G, α, β ∈ Aut(G) > to be the third term of the lower autocentral series of subgroups of G. In this paper, it is shown that every finite abelian group is isomorphic to the third term of the autocentral series of some finite abelian group.

Note Mat. 31(2) (2011), 9 - 16

A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer auto... more A criterion for the existence of groups admitting autocommutator subgroups with cyclic outer automorphism group is given. Also the classification of those finite groups GGG such that K(G)congHK(G)\cong HK(G)congH if HHH is a centerless finite group with cyclic outer automorphism group and possible solutions GGG if ∣Z(H)∣=2|Z(H)|=2∣Z(H)∣=2 and HHH has a cyclic outer automorphism group is presented.

In the present paper we introduce the lower autocentral series of autocommutator subgroups of a g... more In the present paper we introduce the lower autocentral series of autocommutator subgroups of a given group. Following our previous work on the subject in 2009, it is shown that every finite abelian group is isomorphic with n th -term of the lower autocentral series of some finite abelian group.

We compute the autocentral and autoderived series of a fi nite abelian group G corresponding to s... more We compute the autocentral and autoderived series of a fi nite abelian group G corresponding to several groups of automorphisms A. As a result, we determine the autonilpotency and autosolvability of G with respect to A.