Monday, 1 November 2010

Generating Classical Groups over Finite Local Rings

I am interested in classical groups (in particular SLn, Sp2n, SO+n) over finite
rings of the form Rk=mathbbFq[t]/(tk)

for some prime power q (where q is odd in the
orthogonal case) and kinmathbbN. Over a finite field, the maximal subgroups of the
classical groups are known, and so it is for example known that any two semisimple elements of
orders qn+1 and qn1 generate the group Sp2n(mathbbFq) (and similar results for
the other classical groups). I am wondering whether
it is true that any two semisimple elements of orders qn(k1)(qn+1) and qn(k1)(qn1) generate
Sp2n(Rk). Is anyting like this known? Are there lists of maximal subgroups for classical
groups over finite rings? I would also be interested in similar results over mathbbZp/(pk).



EDIT: The answer to the question as asked is usually vacuously yes. Instead, asm meant actually to ask about the group generated by two tori (not just two semisimple elements) of the specified orders, specifically they said (in a now deleted answer):



"I misphrased my question. What I meant to ask is whether any two maximal tori of order qn(k1)(qn+1) and qn(k1)(qn1) generate Sp2n(Rk). Of course these tori are far from cyclic. I guess my head was still in the cyclic case Sp2n(mathbbFq)."

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