International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and Qiangwei Song School of Mathematics and Computation Science Shanxi Normal University Linfen 44, China Copyright c 24 Ligong An and Qiangwei Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we give a transversal for the congruence classes of invertible matrices of order 3 over F 2. Here F 2 denotes the finite field containing 2 elements. Mathematics Subject Classification: 2A5 Keywords: finite field, congruent relation Two matrices A and B over a field F are called congruent if there exists an invertible matrix P over F such that P T AP = B, where T denotes the matrix transpose. Matrix congruence arises when considering the effect of change of basis on Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases. It is a classical result that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues. For matrices over a finite field, we know little about their congruence. In [], a transversal for the congruence classes of invertible matrices of order 2 over F p was given, where F p denotes the finite field containing p elements. This transversal was used to classify some finite p-groups up to isomorphism. In [2], a transversal for the congruence classes of invertible matrices of order 3 over F p was given where p is an odd prime. In this paper, we give a a transversal for the congruence classes of invertible matrices of order 3 over F 2. This work was supported by NSFC (no. 37232& 252), by NSF of Shanxi Province (23-).
24 Ligong An and Qiangwei Song Main Theorem Following matrices form a transversal for the congruence class of invertible matrices of order 3 over F 2. () (2) (3) (4) Proof We use M X to denote X T MX, where X, M G = GL(3, 2). Let φ be a binary function from G G to G such that (M,X) M X. Then φ gives an action of G on G, and two matrices are congruent if and only if they are in the same orbit. So, to prove the theorem, we only need to prove that all orbits are Mi G, where M i is the matrix of Type (i) in the theorem, i =, 2, 3, 4. Firstly, we calculate the order of stabilizers G Mi and the length of orbits Mi G. () We claim that G M = 6 and M G = 28. Let X =(,, ). Then X G M = G E3 if and only if X T X = E 3. That is, T = T = T = and,, are mutually orthogonal. By calculations, {,, } = {(,, ) T, (,, ) T, (,, ) T }. Hence G M = 6 and M G = 28. (2) We claim that G M2 = 3 and M2 G = 56. Let X = M 2 =( T, T, T )M 2. We have X G M2 if and only if X T M 2 X = M 2. That is, =( T, T, T ) + = T + T + T + T. Note that M 2 =(m ij ) satisfies m 2 + m 2 =,m 3 + m 3 = and m 23 + m 32 =, and T + T + T is symmetric. We have T =(n ij ) satisfies n 2 + n 2 =,n 3 + n 3 = and n 23 + n 32 =. Since the rank of T is, T is one of the following six matrices: (a) (d) (b) (e) (c) (f) If T is of Type (a), then =(,, ) and =(,, ). By calculation, T = T + M 2 + T + T =.
Congruence classes of invertible matrices 24 Hence =(,, ) and X = If T is of Type (b), then =(,, ) and =(,, ). By calculation, T = T + M 2 + T + T =. It is a contradiction since T is symmetric. If T is of Type (c), then =(,, ) and =(,, ). By calculation, T = T + M 2 + T + T =. Hence =(,, ) and X = If T is of Type (d), then =(,, ) and =(,, ). By calculation, T = T + M 2 + T + T =. Hence =(,, ) and X = If T is of Type (e), then =(,, ) and =(,, ). By calculation, T = T + M 2 + T + T =. It is a contradiction since T is symmetric. If T is of Type (f), then =(,, ) and =(,, ). By calculation, T = T + M 2 + T + T =.... It is a contradiction since T is symmetric. To sum up, G M2 = 3 and M2 G = 56. (3) We claim that G M3 = 4 and M3 G = 42.
242 Ligong An and Qiangwei Song Let X = M 3 =( T, T, T )M 3. We have X G M3 if and only if X T M 3 X = M 3. That is, =( T, T, T ) + + = T + T + T + T ( + ). Note that M 3 = (m ij ) satisfies m 2 + m 2 =,m 3 + m 3 = and m 23 + m 32 =, and T + T + T is symmetric. We have T ( + ) =(n ij ) satisfies n 2 + n 2 =,n 3 + n 3 = and n 23 + n 32 =. Since the rank of T ( + ) is, T ( + ) is one of the following six matrices: (a) (d) (b) (e) (c) (f) If T ( + ) is of Type (a), then =(,, ) and + =(,, ). By calculation, ( + ) T + T ( + ) = T ( + )+M 3 + T +( + ) T ( + ) =. Hence =(,, ) or (,, ) and X = or. If T ( + ) is of Type (b), then =(,, ) and + =(,, ). By calculation, ( + ) T + T ( + ) = T ( + )+M 3 + T +( + ) T ( + ) = If T ( + ) is of Type (c), then =(,, ) and + =(,, ). By calculation, ( + ) T + T ( + ) = T ( + )+M 3 + T +( + ) T ( + ) =. Hence =(,, ) or (,, ) and X = or..
Congruence classes of invertible matrices 243 If T ( + ) is of Type (d), then =(,, ) and + =(,, ). By calculation, ( + ) T + T ( + ) = T ( + )+M 3 + T +( + ) T ( + ) = If T ( + ) is of Type (e), then =(,, ) and + =(,, ). By calculation, ( + ) T + T ( + ) = T ( + )+M 3 + T +( + ) T ( + ) = If T ( + ) is of Type (f), then =(,, ) and + =(,, ). By calculation, ( + ) T + T ( + ) = T ( + )+M 3 + T +( + ) T ( + ) = To sum up, G M3 = 4 and M G 3 = 42. (4) We claim that G M4 = 4 and M4 G = 42. Let X =. We have X G M4 if and only if X T M 4 X = M 4. That is, M 4 =( T, T, T )M 4 =( T, T, T ) +... = T + T + T + T. Note that M 4 =(m ij ) satisfies m 2 + m 2 =,m 3 + m 3 = and m 23 + m 32 =, and T + T + T is symmetric. We have T =(n ij ) satisfies n 2 + n 2 =,n 3 + n 3 = and n 23 + n 32 =. Since the rank of T is, T is one of the following six matrices: (a) (d) (b) (e) (c) (f)
244 Ligong An and Qiangwei Song If T is of Type (a), then =(,, ) and =(,, ). By calculation, T + T = T + M 4 + T =. If T is of Type (b), then =(,, ) and =(,, ). By calculation, T + T = T + M 4 + T =. If T is of Type (c), then =(,, ) and =(,, ). By calculation, T + T = T + M 4 + T =. If T is of Type (d), then =(,, ) and =(,, ). By calculation, T + T = T + M 4 + T =. Hence =(,, ) or (,, ) and X = or. If T is of Type (e), then =(,, ) and =(,, ). By calculation, T + T = T + M 4 + T = Hence =(,, ) or (,, ) and X = or. If T is of Type (f), then =(,, ) and =(,, ). By calculation, T + T = T + M 4 + T =..
Congruence classes of invertible matrices 245 To sum up, G M4 = 4 and M4 G = 42. Next we prove that M 3 and M 4 are not in the same orbit. If not, then there exists an X = such that X T M 4 X = M 3. That is, M 3 =( T, T, T )M 4 (,, ) T =( T, T, T )( +,,) T = T + T + T + T. Note that M 3 =(m ij ) satisfies m 2 + m 2 =,m 3 + m 3 = and m 23 + m 32 =, and T + T + T is symmetric. We have T =(n ij ) satisfies n 2 + n 2 =,n 3 + n 3 = and n 23 + n 32 =. Since the rank of T is, T is is one of the following six matrices: (a) (d) (b) (e) (c) (f) If T is of Type (a), then =(,, ) and =(,, ). By calculation, T + T = T + M 3 + T =. If T is of Type (b), then =(,, ) and =(,, ). By calculation, T + T = T + M 3 + T =. If T is of Type (c), then =(,, ) and =(,, ). By calculation, T + T = T + M 3 + T =. If T is of Type (d), then =(,, ) and =(,, ). By calculation, T + T = T + M 3 + T =.
246 Ligong An and Qiangwei Song If T is of Type (e), then =(,, ) and =(,, ). By calculation, T + T = T + M 3 + T =. If T is of Type (f), then =(,, ) and =(,, ). By calculation, T + T = T + M 3 + T =. To sum up, there is no X such that X T M 4 X = M 3. Hence M 3 and M 4 are in different obits respectively. Finally, since M G + M 2 G + M 3 G + M 4 G = 68 = G, there is no other orbit. Hence M i form a transversal for the congruence classes of invertible matrices of order 3 over F 2. References [] L.J. An, L.L. Li, H.P. Qu and Q.H. Zhang, Finite p-groups with a minimal nonabelian subgroup of index p (II), Sci. China Math., 57(24), 737 753. [2] H.P. Qu, M.Y. Xu and L.J. An, Finite p-groups with a minimal non-abelian subgroup of index p (III), to appear in Sci. China Math. http://arxiv.org/abs/3.5496. [3] M.Y. Xu, An Introduction to Finite Groups (Chinese), Science Press, Beijing, 987. Received: February 2, 24