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Given a definite quaternion order S over Z, returns a sequence
of representatives of the units in S modulo the unit group of the
base ring.
Given a definite quaternion order S over Z, returns an abstract
group isomorphic to S^ * and the homomorphism into S.
The following example illustrates the unit group computation for an
order in a definite quaternion algebra over Q.
> A := QuaternionAlgebra< RationalField() | -1, -1 >;
> S1 := MaximalOrder(A);
> S2 := QuaternionOrder(A,2);
> G1, h1 := UnitGroup(S1);
> #G1;
24
> [ h1(g) : g in G1 ];
[ 1, -1, i - k, -j + k, 1 - i - j, -j, 1 - k, -1 + i + j - k, -i, i,
1 - i - j + k, j, -1 + k, k, -i - j + k, 1 - j, -1 + i, 1 - i, -1 + j,
-k, i + j - k, j - k, -1 + i + j, -i + k ]
> G2, h2 := UnitGroup(S2);
> #G2;
8
> [ h2(g) : g in G2 ];
[ 1, -1, -i + k, i - k, -1 + i + j, 1 - i - j, -j + k, j - k ]
The unit groups of orders in indefinite quaternion algebras A are
infinite arithmetic groups, which are twisted analogues of the groups
SL_2(Z) and their families of subgroups. These are studied in
relation to their actions on the upper half complex plane, via an
embedding in GL_2(R) provided by some isomorphism A tensor R isomorphic to M_2(R). Currently this perspective is not fully implemented,
but is expected to be part of the generalization of the machinery for
hyperbolic spaces and congruence subgroups.
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