Special linear group of degree two is ambivalent iff -1 is a square

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Statement

Suppose F is a field and G = SL(2,k) is the special linear group of degree two over F. Then, G is an ambivalent group if and only if -1 is a square in F.

In particular, when F is a finite field with q elements, this is equivalent to saying that G is an ambivalent group if and only if q is a power of 2 or q \equiv 1 \pmod 4.

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