Special unitary group of degree two equals special linear group of degree two over a finite field

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Statement

Suppose q is a prime power. Denote by SU(2,q) the special unitary group corresponding to the quadratic extension \mathbb{F}_{q^2} of the finite field \mathbb{F}_q. In particular, SU(2,q) is the intersection of SL(2,q^2) with U(2,q). Then, SU(2,q) is isomorphic to the special linear group of degree two SL(2,q). Moreover, if we view both SU(2,q) and SL(2,q) as subgroups inside SL(2,q^2), they are conjugate subgroups.

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