Defining Relations and the Algebraic Structure of the Group SL2 over Integral Hamilton Quaternions

نویسندگان

Sergei I. Adian

I. G. Lysionok

J. G. Mennicke

چکیده

In the present paper, we study the group SL 2 (H Z) of (2 2)-matrices with reduced norm 1 over the noncommutative ring of Hamilton quaternions with integral coeecients. In section 1, we obtain a presentation for SL 2 (H Z) in terms of generators and deening relations. In section 2, we exhibit a subgroup of nite index of SL 2 (H Z) with a nonabelian free quotient of rank 3. It is well known that the group SL 2 (Z) has the same property. In fact, the commutator subgroup of SL 2 (Z) is a free group of rank 2 and has index 12. The normal subgroup generated in SL 2 (Z) by the element 1 m 0 1 ! (1) coincides with SL 2 (Z) if m = 1, is of nite index if 1 < m 5 and has innnite index otherwise. In SL 2 (H Z), the normal closure of the element (1) also coincides with the whole group if m = 1. But the normal closure of (1) in SL 2 (H Z) turns out to be of innnite index if m = 2. We have tried to keep the exposition of the paper as elementary as possible.

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