editing

proposed

(b) Matrices M with 3 = det(M) mod 8. These ..are the elements of the left coset A*SO(2, Z_8) = {AM: M in SO(2, Z_8)}, where A = [[3,0],[0,1]].

(c) Matrices M with 5 = det(M) mod 8. These ..are the elements of the left coset B*SO(2, Z_8) = {BM: M in SO(2, Z_8)}, where B = [[5,0],[0,1]].

(d) Matrices M with 7 = det(M) mod 8. These ..are the elements of the left coset C*SO(2, Z_8) = {CM: M in SO(2, Z_8)}, where C= [[7,0],[0,1]].

All four classes of matrices have the same number of elements, that is, 16 each. (End)

For n = 2, we list below all the 4*a(2) = 32 64 n X n matrices M with elements in 0..7 that satisfy MM' mod 8 = I can be classified into four categories:

(a) Matrices M with 1 = det(M) mod 8:. These form the abelian group SO(2, Z_8). See the comments for sequence A060968.

These form the abelian group SO(2, Z_8). See the comments for sequence A060968.

(b) Matrices M with 3 = det(M) mod 8:. These ...

(c) Matrices M with 5 = det(M) mod 8:. These ...

(d) Matrices M with 7 = det(M) mod 8:. These ...

From Petros Hadjicostas, Dec 18 2019: (Start)

For n = 2, we list below all 4*a(2) = 32 n X n matrices M with elements in 0..7 that satisfy MM' mod 8 = I:

(a) with 1 = det(M) mod 8:

These form the abelian group SO(2, Z_8). See the comments for sequence A060968.

(b) with 3 = det(M) mod 8:

(c) with 5 = det(M) mod 8:

(d) with 7 = det(M) mod 8:

Jianing Song, <a href="/A060968/a060968.txt">Structure of the group SO(2,Z_n)</a>.

László Tóth, <a href="http://arxiv.org/abs/1404.4214">Counting solutions of quadratic congruences in several variables revisited</a>, arXiv:1404.4214 [math.NT], 2014.

László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Toth/toth12.html">Counting Solutions of Quadratic Congruences in Several Variables Revisited</a>, J. Int. Seq. 17 (2014), #14.11.6.

Cf. A060968, A071302, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083.

1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n x X n identity matrix.

1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n x n identity matrix.

approved

editing

_R. H. Hardin (rhhardin(AT)att.net), _, Jun 12 2002

nonn,new

nonn

Ron R. H. Hardin (rhhardin(AT)att.net), Jun 12 2002