- 983

- 173

I will use a r-dimensional board with n squares along each dimension. A standard chessboard has r = 2 and n = 8. The toroidal-board symmetry group is an affine-symmetry group, a group whose elements are rotation/reflection and translation (R,T), elements that act on a vector x to make vector x' with ##x' = R\cdot x + T##, where x, x', and T are r-vectors, R is an r*r matrix, and all components are in Z

_{n}. Affine symmetry thus generalizes the Euclidean group to components in arbitrary rings, like Z

_{n}under addition and multiplication modulo n. The rings do need additive and multiplicative identities, 0 and 1, for constructing the identity element R = identity matrix and T = zero vector.

For components being in a commutative ring, one can find the inverse of each R by using Cramer's rule. Though impractical for all but the smallest matrices, Cramer's rule is good for theoretical results. It is evident from that algorithm that invertibility requires that each R have a determinant that is a "unit" in that ring, an element with a multiplicative inverse. For ##Z_n##, this is all elements relatively prime to n, or ##Z^\times_n##. It is easy to show that the units of a ring form a group under multiplication.

So I must find the structure of the group of (R,T) for all invertible R and all T over Z

_{n}and number of dimensions r.

The first step is to note that with I the identity matrix, the affine group's subgroup of all (I,T) is a normal subgroup. Its structure is ##(Z_n)^r##, (the component ring's additive group)

^{r}. Its quotient group is the group of R's under matrix multiplication, the homogeneous affine group. This group is ##GL(r,Z_n)##, the general linear group over ring ##Z_n##.

One can proceed further by selecting out all the elements with determinant 1, giving the special linear group: ##SL(r,Z_n)##, The center of the general linear group is all elements ##mI## where m is in ##Z^\times_n##, all the component ring's units times the identity matrix. The quotient group of ##GL(r,Z_n)## and its center is the projective general linear group ##PGL(r,Z_n)##. The special linear group has a similar center, but with the further condition ##m^r = 1##. Its quotient group is the projective special linear group ##PSL(r,Z_n)##.

For r = 1, the group ##GL(1,Z_n) = Z^\times_n##, the units group, and the groups ##SL(1,Z_n)##, ##PGL(1,Z_n)##, and ##PSL(1,Z_n)## are all the identity group.

For n = 2, ##Z^\times_2## is the identity group, and ##GL(r,Z_2) = SL(r,Z_2) = PGL(r,Z_2) = PSL(r,Z_2)##.

One can simplify these groups in another direction, by using a theorem of matrices over rings and their ideals (Combinatorics of Vector Spaces over Finite Fields). A two-sided ideal I of a ring R satisfies ##R\cdot I = I \cdot R = I##, with a left ideal and a right ideal being for only one side. An ideal of the integer ring ##Z## is ##n \cdot Z## for integer n.

The theorem: consider matrix ring ##M(r,C)## of r*r matrices over component ring C. Consider ideals of C, J

_{k}, that are "disjoint". Their intersection ##J = \cap_k J_k##. Then ##M(r,C/J) = \prod_k M(r,C/J_k)##, where each C/J is a quotient ring. This result readily gives analogous results for the GL, SL, PGL, and PSL groups.

Applying to Z

_{n}, that ring is is the quotient ring Z/(nZ), and I've sometimes seen that notation used instead. An appropriate set of disjoint ideals is from prime factors: ##n = \prod_p p^m##, powers m of prime factors p. Thus, ##Z_n = \prod_p Z_{p^m}##. This is also true of the multiplicative group over its units, ##Z^\times_n = \prod_p Z^\times_{p^m}##, of its matrix rings ##M(r,Z_n) = \prod_p M(r,Z_{p^m})##, and of the GL, SL, PGL, and PSL groups.

Multiplicative group of integers modulo n - Wikipedia goes further with ##Z^\times_n = \prod_p Z^\times_{p^m}##, stating the structure of each group ##Z^\times_{p^m}##:

- For p = 2, ##Z^\times_2## is the identity group and ##Z^\times_{2^m} = Z_{2^{m-2}} \times Z_2##.
- For odd p, every other value: ##Z^\times_{p^m} = Z_{p^{m-1}} \times Z_{p-1} = Z_{(p-1)p^{m-1}}##.

Continuing with ##M(r,Z_{p^m})## and its groups, I've only seen stuff on the groups for a related matrix ring, ##M(r,GF(p^m))##, with its component ring being a Galois field. But ##Z_{p^m}## is only a field for m = 1: GF(p).

However, that combinatorics reference states the order of ##GL(r,Z_{p^m})##. It is ##p^{(m-1)r^2} \prod_{k=0}^{r-1} (p^r - p^k)## or ##p^{(m-1)r^2} |GL(r,Z_p)|##. More generally,

$$ |GL(r,Z_n)| = n^{r^2} \prod_p \prod_{k=1}^r \left( 1 - \frac{1}{p^k} \right) $$

over all prime factors p of n. For r = 1, one gets the formula for the Euler totient function. But for Galois fields,

$$ |GL(r,GF(q))| = q^{r^2} \prod_{k=1}^r \left( 1 - \frac{1}{q^k} \right) $$

My question: What has been done on the structure of the group ##PSL(r,Z_{p^m})## for r > 1 and m > 1? For r = 1, it is the identity group. For m = 1, it is simple for all r and m except for r = 2 and m = 2 and 3. That is as far as I can go without doing brute-force calculations.