Let p>3 be a prime and F be a field with p elements. It is clear that S_3 (the symmetric group of order 3!) is a subgroup of SL(2,F). Is there a normal subgroup N of SL(2,F); such that SL(2,F) can be written as a semi-direct product of S_3 with N ?
Tapas Chatterjee
For p=5, this is certainly not true as the only non-trivial normal subgroup of SL(2,5) is its centre which is of order 2.
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Tapas Chatterjee
also for p=7, this is certainly not true as the only non-trivial normal subgroup of SL(2,7) is its centre which is of order 2 as PSL(2,7) is simple.
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