COMMUTING GRAPHS OF MATRIX ALGEBRAS

Transcription

1 Available at: off IC/2006/072 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS COMMUTING GRAPHS OF MATRIX ALGEBRAS S. Akbari 1 Department of Mathematical Sciences, Sharif University of Technology, P.O. Box , Tehran, Iran, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box , Tehran, Iran and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, H. Bidkhori 2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA and A. Mohammadian 3 Department of Mathematical Sciences, Sharif University of Technology, P.O. Box , Tehran, Iran and Institute for Studies in Theoretical Physics and Mathematics, P.O. Box , Tehran, Iran. MIRAMARE TRIESTE August Junior Associate of ICTP. s akbari@sharif.edu 2 hoda bidkhori@yahoo.com 3 ali m@mehr.sharif.edu

2 Abstract The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this paper we investigate some graph theoretic properties of Γ(M n (F)), where F is a field and n 2. Also we study the commuting graphs of some classical groups such as GL n (F) and SL n (F). We show that Γ(M n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL n (F)) and Γ(SL n (F)). Also we show that for two fields E and F and integers m,n 2, if Γ(M m (E)) Γ(M n (F)), then m = n and E = F. 1

3 1. Introduction For a ring R, we denote the center of R by Z(R). If X is either an element or a subset of R, then C R (X) denotes the centralizer of X in R. For any non-commutative ring R, we associate a graph with the vertex set R\Z(R) and join two vertices x and y if and only if x y and xy = yx. This graph was introduced in [2] and is called the commuting graph of R and denoted by Γ(R). If F is a field and n is a natural number, then M n (F) denotes the ring of n n matrices over F, GL n (F) and SL n (F) denote the group of all invertible matrices in M n (F) and the group of all matrices with determinant 1 in M n (F), respectively. In this article, we denote the finite field of order q by F q. For any field F, we set F = F \ {0}, that is the multiplicative group of F. For any i,j, 1 i,j n, we denote by E ij, that element in M n (F) whose (i,j)-entry is 1 and whose other entries are 0. Also, I and I r denote the identity matrix and the identity matrix of size r, respectively. For a matrix A M n (F), F[A] denotes the F-subalgebra generated by A and I. Also for any matrix A M n (F) and α F n, the A-annihilator of α is a polynomial with minimum degree, say f(x), such that f(a)α = 0. The matrix A M n (F) is said to be cyclic if there exists a vector α F n such that {α,aα,...,a n 1 α} is a basis for the vector space F n over F. It is easily checked that a matrix A is cyclic if and only if the minimal and the characteristic polynomials of A coincide. For any two matrices A M n (F) and B M m (F), we define A B = [ A 0 0 B ] M n+m (F). In a graph G, a path P is a sequence of distinct vertices v 1 v 2 v k in which every two consecutive vertices are adjacent. The graph G is called connected if for every pair of vertices u and v in G, there exists a path with endpoints u and v. In this paper, we study the connectivity problem of the graphs Γ(M n (F)), Γ(GL n (F)), and Γ(SL n (F)). Recently, the connectivity problem of the other subgraphs of matrix rings over division rings have been studied in [5, 6]. For further information about the connectivity of the commuting graphs of finite simple groups, see [14]. For a vertex v of a graph G, the degree of v is the cardinality of the set of edges incident with v. A subset X of the vertices of a graph G is called a dominating set if each vertex of G\X is adjacent to at least one vertex of X. The minimum size of a dominating set in a graph G is called the domination number of G and denoted by γ(g). Also a subset Y of the vertices of a graph is called an independent set if the induced subgraph on Y has no edges. The maximum size of an independent set in a graph G is called the independence number of G and denoted by α(g). In [2] it has been proven that for any finite field F and n 2, the following hold. (i) If n < F, then α(γ(m n (F))) ( F 2 + F + 1) F n2 n 2 2. (ii) If n F, then α(γ(m n (F))) ( F 2 + F + 1) F ( F 1) ( n F 2 ) 1. 2

4 Here we improve the lower bound showing that if F is a finite field and n 2, then α(γ(m n (F))) F (n 1)2 +1 ( F 1) n Connectivity of Γ(M n (F)) In [2] it has been shown that for any field F, Γ(M 2 (F)) is not connected and each of its connected components is a complete graph. If F is finite, then the number of connected components of Γ(M 2 (F)) is F 2 + F + 1 and each of them has F 2 F vertices. In this section we prove that for any field F and integer n 3, the graph Γ(M n (F)) is connected if and only if every field extension of F of degree n contains a proper intermediate field. We first recall the following beautiful lemma due to Frobenius. Lemma A. [9, p.111] Let F be a field and n 2. Suppose that A 1 A k is the rational form of a matrix A M n (F). If for each i, n i is the size of the matrix A i and n 1 n k, then dim F C Mn(F)(A) = n 1 + 3n (2k 1)n k. Corollary 1. Let F be a field and n 2. The matrix A M n (F) is cyclic if and only if C Mn(F)(A) = F[A]. Lemma 2. Let F be a field and n 3. If there is a field extension of F of degree n with no proper intermediate fields, then Γ(M n (F)) is not a connected graph. Proof. Let K be a field extension of F with no proper intermediate fields and dim F K = n. Let f(x) F[x] be the minimal polynomial of some element a K \ F and let A M n (F) be the companion matrix of f(x). Clearly, K F[A]. Since A is a cyclic matrix, by Corollary 1 we have C Mn(F)(A) = F[A]. Suppose that B is a non-scalar matrix in F[A]. Since there is no proper intermediate fields between F and K, we conclude that F[B] = F[A]. Thus dim F F[B] = n and B is a cyclic matrix and moreover C Mn(F)(B) = F[B]. This implies that F[A] \ FI is a connected component of Γ(M n (F)). So the proof is complete. The proof of the previous lemma concludes the following corollary. Corollary 3. Let F be a field and n 3. Suppose A is a non-scalar matrix in M n (F) such that F[A] is a field extension of F of degree n with no proper intermediate fields. Then F[A] \ FI is a connected component of Γ(M n (F)). In the next lemma we investigate the existence of a non-cyclic matrix in the F-algebra F[A], where A M n (F). Lemma 4. Let F be a field and n 3 and A M n (F) \ FI. Suppose that F[A] contains at least one proper intermediate field, if F[A] is a field extension of F of degree n. Then F[A] \FI contains a non-cyclic matrix. 3

5 Proof. First suppose that the minimal polynomial of A is reducible. Let f(x) is the minimal polynomial of A and assume that f(x) = g 1 (x)g 2 (x), for two coprime polynomials g 1 (x) and g 2 (x) of degrees at least 1. Using the primary decomposition theorem [8, p.220], we find an integer k and two matrices B 1 M k (F) and B 2 M n k (F) such that the minimal polynomials of B 1,B 2 are g 1 (x),g 2 (x), respectively, and A is conjugate to B 1 B 2. Since n 3, without loss of generality we may assume that k 2. Clearly, A = g 1 (A) has the form 0 C, for some C M n k (F). Now, if ϕ is the minimal polynomial of C, then A ϕ(a ) = 0. This implies that the degree of the minimal polynomial of A is at most n k +1. Thus A is a non-cyclic element of F[A] \ FI. Moreover, if f(x) = p(x) r for some irreducible polynomial p(x) and r 2, then p(a) r/2 is a non-cyclic element contained in F[A] \ FI. Next assume that the minimal polynomial of A is irreducible. So F[A] is a field extension of F. If dim F F[A] < n, then A is not cyclic and so we are done. So suppose that dim F F[A] = n. By the hypothesis, there exists a matrix B F[A] such that FI F[B] F[A]. Since dim F F[B] < n, B is a non-cyclic element of F[A] \ FI. The proof is complete. Lemma 5. Let F be a field and n 3. Then for every two non-cyclic matrices X and Y in M n (F), there exists a path with non-invertible intermediate vertices between X and Y in Γ(M n (F)). Proof. Assume that A is a non-cyclic matrix in M n (F). Clearly, it is enough to show that there exists a path with non-invertible intermediate vertices between A and E 11. By Lemma A, we have dim F C Mn(F)(A) n + 1. Let {C 1,...,C n+1 } be a linearly independent subset of C Mn(F)(A), where C 1 = I. There exist scalars λ 1,...,λ n+1 such that the matrix A 1 = n+1 i=1 λ ic i is a non-zero matrix whose first row is 0. Suppose that A 1 = X B. Let g(x) = x m + b m 1 x m b 0 be the B-annihilator of X. First assume that g(x) is not equal to the minimal polynomial of B. It is easy to check that both A 1 and E 11 commute with A 2 = 0 1 g(b). Hence A A 1 A 2 E 11 is a path in Γ(M n (F)) with non-invertible intermediate vertices, as desired. Next suppose that g(x) is the minimal polynomial of B. We have g(a 1 ) = b Y 0 n 1, where Y = (B m 1 + b m 1 B m b 1 I)X. Since Y 0, we have g(a 1 ) 0. Moreover there exists a non-zero matrix B M n 1 (F) such that B Y = 0. Now obviously the matrix A 2 = 0 1 B commutes with g(a 1 ). Therefore A A 1 g(a 1 ) A 2 E 11 is a path in Γ(M n (F)) with non-invertible intermediate vertices. The proof of the lemma is complete. 4

6 Now we are in a position to prove the main theorem of this section. Theorem 6. Let F be a field and n 3. The graph Γ(M n (F)) is connected if and only if every field extension of F of degree n contains at least a proper intermediate field. Proof. By Lemma 2, one direction is clear. For other direction, consider two vertices A and B in Γ(M n (F)). By Lemma 4, we find two non-cyclic matrices A C Mn(F)(A) and B C Mn(F)(B). Now, using Lemma 5, there exists a path between A and B in Γ(M n (F)). This completes the proof. Corollary 7. Suppose that F is a finite field and n 2. The graph Γ(M n (F)) is connected if and only if n is not a prime number. Remark 8. In [11] it has been shown that for any n, there is a field extension of Q of degree n with no proper intermediate fields, where Q is the field of rational numbers. Thus for each n 2, the graph Γ(M n (Q)) is not connected. Moreover, if F is either the field of real numbers or an algebraically closed field, then for n 3, Γ(M n (F)) is a connected graph. 3. Connectivity in Γ(GL n (F)) In this section we would like to determine under which conditions does there exist a path with invertible intermediate vertices between two vertices of Γ(M n (F)). The following lemma is used in our proofs frequently. Lemma B. [13, Theorem ] Suppose that A M r (F) and B M s (F) are two matrices such that the minimal polynomials of A and B are coprime. Then C Mn(F)(A B) = {X Y X C Mr(F)(A) and Y C Ms(F)(B), where n = r + s}. Theorem 9. Let F be a field and n 3. Suppose that two vertices X and Y are contained in the same connected component of Γ(M n (F)). Then there is a path with invertible intermediate vertices between X and Y, unless n 1 is a prime number and F = F 2. Proof. Clearly, it suffices to show that for any two non-adjacent vertices X and Y, if X A Y is a path in Γ(M n (F)), then there are invertible matrices P 1,...,P l such that X P 1 P l Y is a path in Γ(M n (F)). If there is an invertible matrix P F[A] \ FI, then there is nothing to prove. So assume that all invertible matrices in F[A] are scalar matrices. This implies that F[A] has no non-zero nilpotent elements. Therefore the minimal polynomial of A is the product of some distinct irreducible polynomials. By the primary decomposition theorem [8, p. 220] and 5

7 with no loss of generality, we may write A = A 1 A r, where the minimal polynomials of matrices A 1,...,A r are irreducible and distinct. Hence F[A 1 ],...,F[A r ] are fields and F[A] F[A 1 ] F[A r ]. Since each unit of F[A] is a scalar matrix, F[A 1 ],...,F[A r ] are trivial groups. This yields that F = F 2 and A is an idempotent matrix. With no loss of generality suppose that A = I k 0 n k, for some k. Since X and Y commute with A, we have X = X 1 X 2 and Y = Y 1 Y 2, where X 1,Y 1 M k (F) and X 2,Y 2 M n k (F). Assume that n 1 is not a prime number. To complete the proof, we consider two cases as follows. Case 1. 2 k n 2. We claim that for any matrix S M m (F) with m 3, there is a matrix S M m (F) which is not idempotent and SS = S S. If S is not idempotent, then put S = S. Suppose that S is an idempotent matrix. There is an invertible matrix Q such that QSQ 1 = I s 0 m s, where s = ranks. If s 2, then put S = Q 1 E 12 Q; otherwise m s 2 and in this case set S = Q 1 E m, m 1 Q. So the claim is proved. Since n 1 is not a prime number, we have n 5. If k 3, then there are two matrices X 1 M k (F) and Y 1 M k(f) which are not idempotent and commute with X 1 and Y 1, respectively. Now since n k 2, X X 1 I n k (I n + E n, n 1 ) Y 1 I n k Y is a path in Γ(M n (F)). Furthermore, since the matrices X 1 I n k and Y 1 I n k are not idempotent, by the first step of the proof there are two invertible matrices P 1 F[X 1 I n k]\fi and P 2 F[Y 1 I n k] \FI. Thus X P 1 (I n +E n, n 1 ) P 2 Y is a path with invertible intermediate vertices. If k = 2, then n k 3 and a similar argument works. Case 2. k = 1 or k = n 1. First assume that k = 1. Since XY Y X, X 2 and Y 2 are not scalar matrices and so they are two vertices of Γ(M n 1 (F)). Note that n 1 is not a prime number and F = F 2, so by Corollary 7 there is a path X 2 B 1 B t Y 2 in Γ(M n 1 (F)). Now, X λ 1 I 1 B 1 λ t I 1 B t Y ( ) is a path in Γ(M n (F)), with λ i = 0 if B i is an idempotent matrix of rank at least 2, and otherwise λ i = 1. For an index j, if B j is idempotent, then rankb j n 1, since B j I n 1. Thus we have 2 rank(λ j I 1 B j ) n 2. Therefore, by Case 1, we can replace the vertex λ j I 1 B j in ( ) with a path whose vertices are invertible matrix. Also if B j is not an idempotent matrix, then F[λ j I 1 B j ] \ FI contains an invertible matrix. Hence in this case we also find a path between X and Y whose intermediate vertices are invertible. The case k = n 1 is proven similarly. So the proof is complete. In the next remark we will show that if n 1 is a prime number, then Γ(GL n (F 2 )) is not connected. Corollary 10. Suppose that F F 2 is a field and n 3. The graph Γ(GL n (F)) is connected if and only if every field extension of F of degree n contains a proper intermediate field. 6

8 Proof. By Theorems 6 and 9, one direction is clear. Assume that there is a field extension of F of degree n with no proper intermediate fields. Using Corollary 3, there is an invertible matrix A such that F[A] \ FI is a connected component of Γ(M n (F)) and each matrix contained in F[A] \ FI is cyclic. Now, B = I + E 12 is an invertible matrix which is not cyclic and there is no path between A and B. So Γ(GL n (F)) is not a connected graph. This completes the proof. Combining Theorem 6 and Corollary 10 and using [2, Theorem 2], we obtain the following result. Corollary 11. Suppose that F F 2 is a field and n 2. The graph Γ(GL n (F)) is connected if and only if the graph Γ(M n (F)) is connected. 4. Connectivity in Γ(SL n (F)) In this section we determine whether two vertices of Γ(M n (F)) are connected by a path whose intermediate vertices have determinant 1. Lemma C. [12, Corollary16.4.b] Let F be a field and n be a natural number. For any matrix A GL n (F), there exist matrix C SL n (F) such that A n = (det A)C. Theorem 12. Let F be a finite field and n 3. Suppose that either n 1 is not a prime number or n is not divisible by F 1. Then for any two vertices X and Y which are contained in the same connected component of Γ(M n (F)), there is a path between X and Y whose intermediate vertices have determinant 1. Proof. Clearly, it suffices to prove that for any two non-adjacent vertices X and Y, if X A Y is a path in Γ(M n (F)), then there are matrices U 1,...,U s SL n (F) such that X U 1 U s Y is a path in Γ(M n (F)). If there is a non-zero nilpotent matrix N F[A], then I + N C Mn(F)({X,Y }) has determinant 1 and so there is nothing to prove. So we may assume that F[A] has no non-zero nilpotent element. Therefore the minimal polynomial of A is the product of some distinct irreducible polynomials. By the primary decomposition theorem [8, p.220] and with no loss of generality, we may write A = A 1 A k, where the minimal polynomials of matrices A 1,...,A k are irreducible and distinct. Since X and Y commute with A, by Lemma B we have X = X 1 X k and Y = Y 1 Y k, where for i = 1,...,k, the matrices X i and Y i commute with A i. We consider the following two cases. Case 1. With no loss of generality, assume that A 1 is not a scalar matrix. Since the minimal polynomial of A 1 is irreducible, F[A 1 ] is a finite field. Let r be the size of A 1 and m = dim F F[A 1 ]. Therefore the order of the multiplicative cyclic group F(A 1 ) /F is F m F + 1 which is more than m. Thus there is a matrix B F[A 1 ] such that B m is not a scalar 7

9 matrix. Assume that A 1 = A 0 A 0 is the rational form of A 1, where A 0 is a cyclic matrix of size m. Since B F[A 1 ], we can write B = B 0 B 0, for some B 0 M m (F). Now B m 0 is not a scalar matrix, so using Lemma C we find a matrix C 0 F[B 0 ] \FI with determinant 1. Set C = C 0 C 0, where the number of C 0 and the number of A 0 appearing in the rational form of A 1 are the same. Now, since C F[A 1 ], C I n r is a matrix with determinant 1 in C Mn(F)({X,Y }). So in this case, we are done. Case 2. Suppose that A 1,...,A k are scalar matrices. If k = n, then since A i A j, for i j, X and Y are diagonal matrices and so XY = Y X, a contradiction. Thus k < n and without loss of generality we may assume that r, the size of A 1, is more than 1. If F = F 2, then the result follows from Theorem 9. So assume that F F 2. Assume first that r n 2. If there exists a matrix P GL r (F) such that PX 1 P 1 is diagonal, then define X = P 1 diag(λ 2,λ r n 2,1,...,1)P λi n r, for some λ 0,1. Otherwise, X 1 is not diagonalizable and so by Case 1 there is a non-scalar matrix Q SL r (F) which commutes with X 1. In this case set X = Q I n r. Similarly define a non-scalar matrix Y SL n (F) correspondence to Y 2 Y k and note that X X Y Y is a path whose intermediate vertices have determinant 1. So the assertion is proved. Now, assume that r = n 1. If n is not divisible by F 1, then there is a scalar µ F with µ n 1. Therefore X µi r µ 1 n I 1 Y is a path in Γ(M n (F)), as desired. So assume that n 1 is not a prime number. Since XY Y X, the matrices X 1 and Y 1 are not scalars and so they are two vertices in Γ(M n 1 (F)). Then by Theorem 9 we find a path X 1 P 1 P t Y 1 in Γ(M n 1 (F)), where P i GL n 1 (F) for i = 1,...,t. Now, X P 1 (det P 1 ) 1 I 1 P t (det P t ) 1 I 1 Y is a path in Γ(M n (F)) whose intermediate vertices have determinant 1. This completes the proof. Remark 13. Note that the converse of the previous theorem is also true. Assume that n 1 is a prime number and F 1 divides n. Let S 0 M n 1 (F) be a cyclic matrix such that its minimal polynomial is irreducible and put S = S 0 (det S 0 ) 1 I 1. By Corollary 1 and Lemma B, every vertex adjacent to S has the form Z νi 1, ( ) where Z F[S 0 ] and ν F. Suppose that T is a vertex not adjacent to S and S Z 0 ν 0 I 1 T is a path in Γ(M n (F)). We show that Z 0 is a scalar matrix. Working towards a contradiction, assume that Z 0 FI. Since F[S 0 ] is a field with prime degree over F and Z 0 F[S 0 ], we have F[Z 0 ] = F[S 0 ]. This implies that Z 0 is a cyclic matrix and therefore every vertex adjacent to Z 0 ν 0 I 1 has the form ( ). But this is a contradiction, since T is not adjacent to S. Hence every path in Γ(M n (F)) between S and a vertex which is not adjacent to S, contains a vertex of the form αi n 1 βi 1, for some α,β F. Now, since the nth powers of all elements of F are 8

10 equal to 1, every matrix of the form αi n 1 βi 1 in SL n (F) is scalar. Hence there is no path between S and T whose intermediate vertices have determinant 1. Corollary 14. Suppose that F is a finite field and n 2. The graph Γ(SL n (F)) is not a connected graph if and only if at least one of the following cases occurs. (i) n is a prime number. (ii) n 1 is a prime number and F 1 divides n. Proof. Suppose first that for two vertices X and Y in Γ(SL n (F)), there is no path between X and Y in Γ(SL n (F)). If there is no path between X and Y in Γ(M n (F)), then by Corollary 7, Case (i) occurs. Otherwise, by Theorem 12, Case (ii) occurs. For the other direction, assume first that n is a prime number. Let K be a field extension of F of degree n. By [3, Corollary1], there is an element a K \ F such that N K/F (a) = 1. Consider the map L a : K K defined by L a (x) = ax. Thus the matrix representation of L a, say A, is contained in SL n (F) \ FI. Hence A is a cyclic matrix with determinant 1 and dim F F[A] is a prime number. Now, using Corollary 3, we conclude that F[A] SL n (F) \ FI is a connected component of Γ(SL n (F)) and so this graph is not connected. Moreover, if n 1 is a prime number and F 1 divides n, then by Remark 13 we conclude that Γ(SL n (F)) is not connected. Theorem 15. Let F F 2 be a field and n 2. For every matrix A M n (F), the set F[A]\FI contains a matrix with determinant 1, unless F[A] is a field, F[A]/F is purely inseparable, and dim F F[A] divides n. Furthermore, if F has a purely inseparable field extension and char F divides n, then there exists a matrix B M n (F) such that none of the elements of F[B] \ FI has determinant 1. Proof. If there is a non-zero nilpotent matrix N F[A], then I + N has determinant 1 and so there is nothing to prove. Thus we may assume that F[A] is a reduced ring. Therefore there are fields K 1,...,K r such that F[A] K 1 K r. If r 2, then the element of F[A] corresponding to the element (λ,λ 1,1,...,1), for some λ F \ {0,1}, is a non-scalar matrix with determinant 1. So assume that r = 1. If F is finite, then using [3, Corollary 1], we can find an element of F[A] \ FI with determinant 1. Hence suppose that F is infinite. If there exists a matrix X F[A] such that X n is not a scalar matrix, then by Lemma C, there exists a matrix C F[A] \FI with determinant 1. So assume that F[A] /F is an Abelian group whose exponent divides n. Now [1, Lemma13] implies that F[A] is a purely inseparable extension of F. Let char F = p. Since F[A]/F is a purely inseparable field extension, there is an integer s such that dim F F[A] = p s. Therefore the order of AF in the group F[A] /F is p s. This implies that dim F F[A] divides n, as desired. 9

11 To complete the proof, suppose that F has a purely inseparable field extension K and p = char F divides n. Without loss of generality, we may suppose that dim F K = p. Let E be the subfield of F generated by all algebraic elements over F p. Suppose that ω K \ F and x p t is the minimal polynomial of ω over F. Since K/F is a purely inseparable field extension, t is not an algebraic element over E. Let S F be a transcendental basis of F over E which contains t. Let F 0 = E(S \ {t}), so we have F = F 0 (t) and ω is transcendental over F 0. Now, let T be the companion matrix of the polynomial x p t. Put B = T T, where the number of T is n/p. We show that none of the elements of F[B] \ FI has determinant 1. Let Y = λ 1 B p λ p 1 B + λ p I be an arbitrary element of F[B] \ FI. Also put Y 0 = λ 1 T p λ p 1 T + λ p I. Since p = char F and T p = t, Y p 0 (λp 1 tp λ p p 1 t + λp p)i = 0. ( ) This implies that the minimal polynomial of Y 0 as a matrix over F has the form (x α) k, where F is the algebraic closure of F and k 2. Thus the characteristic polynomial of Y 0 is (x α) p. Now, using ( ), we find that x p (λ p 1 tp λ p p 1 t + λp p) is the characteristic polynomial of Y 0. So if det Y = 1, then (λ p 1 tp 1 + +λ p p 1 t+λp p) n/p = 1 and (λ 1 ω p 1 + +λ p 1 ω+λ p ) n = 1. On the other hand λ 1,...,λ p F = F 0 (ω p ). So the latter equation shows that ω is an algebraic element over F 0, a contradiction. Hence the proof is complete. Theorem 16. Let F be an infinite field and n 2. Suppose that two non-adjacent vertices A and B are contained in the same connected component of Γ(M n (F)). Also suppose that F does not have a purely inseparable extension of degree n. Then there exists a path between A and B whose intermediate vertices have determinant 1. Proof. If F[A] is a field extension of F of degree n with no proper intermediate fields, then using Corollary 3, F[A] \FI is a connected component of Γ(M n (F)). Therefore the two vertices A and B are adjacent, a contradiction. So we may assume that if F[A] is a field of degree n, then there is at least one field between F and F[A]. Clearly, it is sufficient to find a path between A and M = λ 1 n I 1 λi n 1 whose intermediate vertices have determinant 1, where λ is a scalar such that λ n 0,1. Case 1. If either F[A] is not a field or A is a non-cyclic matrix, then by Lemmas 4 and 5, there is a path between A and M whose intermediate vertices are non-invertible matrices. Now, using Theorem 15, we can replace intermediate vertices of this path with vertices which have determinant 1. Case 2. Assume that A is a cyclic matrix and F[A] is a field of degree n. So there is a matrix A 0 F[A] such that F[A 0 ] F[A] and F[A 0 ] is a separable extension of F. Hence by Theorem 15, there is a matrix A 1 F[A 0 ] \ FI with determinant 1. Now since A 1 is not a cyclic matrix, using Case 1 we find a path between A 1 and M whose intermediate vertices have determinant 1. 10

12 Corollary 17. Suppose that F is an infinite field and n 3. If every field extension of F of degree n contains at least a proper intermediate field, then Γ(SL n (F)) is a connected graph. Proof. Using Case 1 in the proof of Theorem 16, it sufficient to show that any cyclic matrix in SL n (F) is adjacent to at least a non-cyclic matrix in SL n (F). Assume that A SL n (F) is a cyclic matrix. If F[A] is not a field, then by Lemma 4, there is a non-cyclic matrix B F[A]\FI which is not invertible. Therefore by Theorem 15, there is a matrix B 1 F[B] \ FI with determinant 1. Since B 1 SL n (F) is a non-cyclic matrix which commutes with A, we are done. Now, suppose that F[A] is a field. Working towards a contradiction, assume that F[A] is a purely inseparable extension of F. Hence the minimal polynomial of A is x n + ( 1) n. Since n is a power of char F, (A I) n = 0. On the other hand, F[A] is a field, so we have A = I, a contradiction. Thus there exists a matrix C F[A] such that F[C] F[A] and F[C] is a separable extension of F. Hence by Theorem 15, there exists a matrix C 1 F[C] \ FI with determinant 1. Now, C 1 SL n (F) is a non-cyclic matrix commuting with A and the proof is complete. 5. Some graph theoretic parameters of commuting graphs In this section we first improve the lower bound for α(γ(m n (F))) given in [2] and show that if F is a finite field and n 2, then α(γ(m n (F))) F (n 1)2 +1 ( F 1) n 2. We begin with the following lemma. Lemma D. [7, p.91] Let G be a graph with n vertices. If d 1,...,d n are the degrees of all vertices of G, then α(g) n i=1 1 d i +1. Theorem 18. If F is a finite field and n 2, then α(γ(m n (F))) F (n 1)2 +1 ( F 1) n 2. Proof. If A is a cyclic matrix in M n (F), then by Corollary 1 the degree of A in Γ(M n (F)) is F n F 1. Let A 1,...,A m be all cyclic matrices in M n (F) with distinct minimal polynomials. Clearly, we have m = F n. Moreover, if A M n (F) is a cyclic matrix, then using Corollary 1, there are at most F n 1 invertible matrices commuting with A. Thus the number of conjugates of A is at least GL n (F) /( F n 1). Therefore the total number of cyclic matrices is at least t = F n GL n (F) /( F n 1). By Lemma D, we conclude that α(γ(m n (F))) t i=1 Thus we find that α(γ(m n (F))) This implies that F n ( F n F )( F n 1) n 1 n 1 ( F n F i ) = F n ( F n F i ). α(γ(m n (F))) F n ( F n F n 1 ) n 2 = F (n 1)2 +1 ( F 1) n 2. i=0 i=2 1 F n F. So the proof is complete. 11

13 Now we state a result about the domination numbers of commuting graphs. Theorem 19. Let F be an infinite field and n 2. Then γ(γ(m n (F))), γ(γ(gl n (F))), and γ(γ(sl n (F))) are infinite. Proof. It is well known that every vector space over an infinite field cannot be a union of finitely many its proper subspaces. Thus for any vertices A 1,...,A k of Γ(M n (F)), we have M n (F) C Mn(F)(A 1 ) C Mn(F)(A k ). Clearly, this implies that γ(γ(m n (F))) is infinite. Now, assume thaty is a dominating set for Γ(GL n (F)). Since F is infinite, for any matrix A M n (F), there exists a scalar α F such that αi A is invertible. Therefore there exists a vertex X Y such that X commutes with αi A. Thus if A X, then A X is an edge of Γ(M n (F)). This yields thaty is also a dominating set for Γ(M n (F)). So γ(γ(gl n (F))) is infinite. Finally, we prove that γ(γ(sl n (F))) is infinite. To get a contradiction, suppose {S 1,...,S m } is a finite dominating set for Γ(SL n (F)). We have SL n (F) = C SLn(F)(S 1 ) C SLn(F)(S m ). It is well known that Z(SL n (F)) FI, so C SLn(F)(S i ) is a proper subgroup of SL n (F), for i = 1,...,m. By Neumann s Lemma [10, p.92], at least one of these subgroups is of finite index in SL n (F). Thus it is easy to see that SL n (F) contains a normal subgroup N of finite index r. On the other hand, by [4, Theorem 11], we have N FI. Hence for any λ F, diag(λ,λ 1,1,...,1) r is a scalar matrix. This implies that λ 2r = 1, for each λ F, a contradiction. Now, the proof is complete. 6. A uniqueness theorem for commuting graphs In [2] it has been proved that if E and F are two finite fields such that Γ(M m (E)) Γ(M n (F)), m,n 2, then m = n and E = F. In the sequel, we will generalize this result for arbitrary fields. Lemma E. [2, Lemma2] Let F be a field, n 1, and A,B M n (F). If C Mn(F)(A) C Mn(F)(B), then there exists a polynomial f(x) F[x] such that B = f(a). Lemma 20. Let F be a field and n 1. If C Mn(F)(A 1 ) C Mn(F)(A k ) is a chain of centralizers, then k n. Moreover, if F n, then there exists a chain of distinct centralizers in M n (F) of length n. Proof. Suppose that C Mn(F)(A) C Mn(F)(B), for A,B M n (F). Then by Lemma E, we have B F[A]. Since C Mn(F)(A) C Mn(F)(B), we conclude that F[B] F[A]. Hence the degree of the minimal polynomial of B is less than the degree of the minimal polynomial of A. Now, if C Mn(F)(A 1 ) C Mn(F)(A k ) is a chain and m i is the degree of the minimal 12

14 polynomial of A i, then n m 1 > > m k 1. This implies that k n. Furthermore, if a 1,...,a n are distinct elements of F, then the following chain C Mn(F)( diag(a1,a 2,...,a n ) ) ( C Mn(F) diag(a2,a 2,a 3,...,a n ) ) ( C Mn(F) diag(an 1,...,a n 1,a n ) ) ( C Mn(F) diag(an,...,a n ) ) has length n. This completes the proof. The uniqueness of the Wedderburn-Artin theorem states that if E and F are two fields and M m (E) M n (F), then m = n and E F. In the next theorem we prove a similar uniqueness theorem for the commuting graphs. Theorem 21. If E and F are two fields, m,n 2 and Γ(M m (E)) Γ(M n (F)), then m = n and E = F. Proof. By [2, Corollary 2], we may assume that E and F are infinite. Since M m (E) \ EI is the vertex set of Γ(M m (E)) and E is infinite, the cardinal of the vertex set of Γ(M m (E)) is E. Now, since Γ(M m (E)) Γ(M n (F)), we have E = F. To complete the proof, for any vertex A of Γ(M m (E)), let N(A) be the set of all vertices X in Γ(M m (E)) such that either X = A or X A is an edge of Γ(M m (E)). Assume that N(A 1 ) N(A k ) ( ) is a chain in Γ(M m (E)) with maximum possible length. Then clearly, C Mm(E)(A 1 ) C Mm(E)(A k ) C Mm(E)(I) is a chain of centralizers in M m (E) with maximum length. By Lemma 20, we conclude that k = m 1. Similarly, the size of every chain of type ( ) with maximum length in Γ(M n (F)) is n 1. Since Γ(M m (E)) Γ(M n (F)), we have m = n, as desired. Acknowledgments The first and third authors are indebted to the Research Council of Sharif University of Technology and the Institute for Studies in Theoretical Physics and Mathematics (IPM) for support. The research of the first author was in part supported by a grant (No ) from IPM. This work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. 13

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