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

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Statement

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

In particular, when k 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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