Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

On <Emphasis Type="Italic">n</Emphasis>-commuting and <Emphasis Type="Italic">n</Emphasis>-skew-commuting maps with generalized derivations in prime and semiprime rings

Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h: R → R is called n-centralizing (n-commuting) on a subset S of R if [h(x),x ⁿ ] ∈ Z(R) ([h(x),x ⁿ ] = 0 respectively) for all x ∈ S. The following are proved:

An Engel condition with generalized derivations on multilinear polynomials

Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x ₁,..., x _n) a multilinear polynomial over C, I a nonzero right ideal of R. If [g(f(r ₁,..., r _n)), f(r ₁,..., r _n)] = 0, for all r ₁, ..., r _n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

On generalized derivations satisfying certain identities

Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities:

Recognition by noncommuting graph of finite simple groups L 4(q)

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x, y] ≠ 1. We prove that if L ∈ {L ₄(7), L ₄(11), L ₄(13), L ₄(16), L ₄(17)} and G is a finite group such that ∇(G) ≅ ∇(L), then G ≅ L.     

On the Cauchy difference on normed spaces

LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K.     

On uniformly continuous Nemytskij operators generated by set-valued functions

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all Hölder functions ${\varphi : I \to C}Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all H?lder functions j: I ? C{\varphi : I \to C} into the set H _β (I, clb(Z)) of all H?lder set-valued functions f: I ? clb(Z){\phi : I \to clb(Z)} and is uniformly continuous, then

F(x,y)=A(x,y) \text+* B(x),       x ? I, y ? CF(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C      10. A Generalization of Orthogonal Factorizations in Graphs      Guo Jun Li  Gui Zhen Liu 《数学学报(英文版)》2001,17(4):669-678 Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let = {F 1, F 2, ..., F m } be a factorization of G and H be a subgraph of G with mr edges. If F i , 1 ≤i≤m, has exactly r edges in common with H, then is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges. This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China      11. Derivations with power central values on multilinear polynomials in prime rings      Basudeb Dhara 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):401-410 Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x 1, ..., x n ) a nonzero multilinear polynomial over C. Suppose that x s d(x)x t ∈ Z(R) for all x ∈ {d(f(x 1, ..., x n ))|x 1, ..., x n ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ C = eRC for some idempotent e in the socle of RC and f(x 1, ..., x n ) N is central-valued in eRCe, where N = s + t + 1.       12. An Application of a Mountain Pass Theorem   总被引：3，自引：0，他引：3   Huan?Song?ZhouEmail author 《数学学报(英文版)》2002,18(1):27-36 We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H 1 0(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ s 0 f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.      13. Strict Betweennesses Induced by Posets as well as by Graphs      Dieter Rautenbach  Philipp Matthias Schäfer 《Order》2011,28(1):89-97 For a finite poset P = (V, ≤ ), let _s(P){\cal B}_s(P) consist of all triples (x,y,z) ∈ V 3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let Bs(G){\cal B}_s(G) consist of all triples (x,y,z) ∈ V 3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations Bs(P){\cal B}_s(P) and Bs(G){\cal B}_s(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation Bs(P)=Bs(G){\cal B}_s(P)={\cal B}_s(G).      14. A descent method for submodular function minimization      Satoru Fujishige  Satoru Iwata 《Mathematical Programming》2002,92(2):387-390 We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 V with ?,V∈? and f(?)=0 and for any vector x∈R V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z 1,Z 2,···,Z k of V such that f(Z 1)>f(Z 2)>···>f(Z k )=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001      15. Functional equations on abelian groups with involution      H. Stetkær 《Aequationes Mathematicae》1997,54(1-2):144-172 Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ I 2 =1 g l (x)h l (y),x, y∈G, where the functionsf,g 1,h 1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.      16. Factors and Connected Induced Subgraphs      Keiko Kotani 《Graphs and Combinatorics》2001,17(3):511-515  Let G be a connected graph without loops and without multiple edges, and let p be an integer such that 0 < p<|V(G)|. Let f be an integer-valued function on V(G) such that 2≤f(x)≤ deg G (x) for all x∈V(G). We show that if every connected induced subgraph of order p of G has an f-factor, then G has an f-factor, unless ∑ x ∈ V ( G ) f(x) is odd. Received: June 29, 1998?Final version received: July 30, 1999      17. Existence of subgraph with orthogonal (g,f)-factorization      Yan Guiying  Pan Jiaofeng 《中国科学A辑(英文版)》1998,41(1):48-54 Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F 1,F 2,⃛,F 1| ofG, if |E(H)∩E(F 1)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.      18. Generalized derivations with Engel condition on multilinear polynomials      Vincenzo De Filippis 《Israel Journal of Mathematics》2009,171(1):325-348 Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x 1, ..., x n ) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer. If [δ(f(r 1, ..., r n )), f(r 1, ..., r n )] k = 0, for all r 1, ..., r n ∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds (1) if char(R) = 0 then f(x 1, ..., x n ) is central valued in eRCe (2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s 4 (3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x 1, ..., x n )2 is central valued in eRCe.      19. Separating maps on weighted function algebras on topological groups      Saeid Maghsoudi  Rasoul Nasr-Isfahani 《Mathematica Slovaca》2011,61(6):941-948 Let G 1 and G 2 be locally compact groups and let ω 1 and ω 2 be weight functions on G 1 and G 2, respectively. For i = 1, 2, let also C 0(G i , 1/ω i ) be the algebra of all continuous complex-valued functions f on G i such that f/ω i vanish at infinity, and let H: C 0(G 1, 1/ω 1) → C 0(G 2, 1/ω 2) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C 0(G 1, 1/ω 1) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C 0(G 1, 1/ω 1), where φ: G 2 → ℂ is a non-vanishing continuous function and h: G 2 → G 1 is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.      20. Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>      Công-Trình Lê  Trung-Hiêu Thái 《Aequationes Mathematicae》2011,82(3):269-276 Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy −1) =  2f(x) and f(xy) + f(y −1 x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GLn(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.

A Generalization of Orthogonal Factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d _H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let = {F ₁, F ₂, ..., F _m } be a factorization of G and H be a subgraph of G with mr edges. If F _i , 1 ≤i≤m, has exactly r edges in common with H, then is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges. This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China     

Derivations with power central values on multilinear polynomials in prime rings

Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C. Suppose that x ^s d(x)x ^t ∈ Z(R) for all x ∈ {d(f(x ₁, ..., x _n ))|x ₁, ..., x _n ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ C = eRC for some idempotent e in the socle of RC and f(x ₁, ..., x _n ) ^N is central-valued in eRCe, where N = s + t + 1.      

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Strict Betweennesses Induced by Posets as well as by Graphs

For a finite poset P = (V, ≤ ), let _s(P){\cal B}_s(P) consist of all triples (x,y,z) ∈ V ³ such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let B_s(G){\cal B}_s(G) consist of all triples (x,y,z) ∈ V ³ such that y is an internal vertex on an induced path in G between x and z. The ternary relations B_s(P){\cal B}_s(P) and B_s(G){\cal B}_s(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation B_s(P)=B_s(G){\cal B}_s(P)={\cal B}_s(G).     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Functional equations on abelian groups with involution

Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ _I ² =₁ g _l (x)h _l (y),x, y∈G, where the functionsf,g ₁,h ₁ to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.     

Factors and Connected Induced Subgraphs

 Let G be a connected graph without loops and without multiple edges, and let p be an integer such that 0 < p<|V(G)|. Let f be an integer-valued function on V(G) such that 2≤f(x)≤ deg _G (x) for all x∈V(G). We show that if every connected induced subgraph of order p of G has an f-factor, then G has an f-factor, unless ∑ _{x ∈ V ( G )} f(x) is odd. Received: June 29, 1998?Final version received: July 30, 1999     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Generalized derivations with Engel condition on multilinear polynomials

Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer. If [δ(f(r ₁, ..., r _n )), f(r ₁, ..., r _n )] _k = 0, for all r ₁, ..., r _n ∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds (1) if char(R) = 0 then f(x ₁, ..., x _n ) is central valued in eRCe (2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s ₄ (3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x ₁, ..., x _n )² is central valued in eRCe.     

Separating maps on weighted function algebras on topological groups

Let G ₁ and G ₂ be locally compact groups and let ω ₁ and ω ₂ be weight functions on G ₁ and G ₂, respectively. For i = 1, 2, let also C ₀(G _i , 1/ω _i ) be the algebra of all continuous complex-valued functions f on G _i such that f/ω _i vanish at infinity, and let H: C ₀(G ₁, 1/ω ₁) → C ₀(G ₂, 1/ω ₂) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C ₀(G ₁, 1/ω ₁) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C ₀(G ₁, 1/ω ₁), where φ: G ₂ → ℂ is a non-vanishing continuous function and h: G ₂ → G ₁ is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.     

Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy ⁻¹) =  2f(x) and f(xy) + f(y ⁻¹ x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GL_n(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S _n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S _n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.