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A normal subgroup satisfies $gH = Hg$ because its defining property is that it is invariant under conjugation, meaning $gHg^{-1} = H$. By right-multiplying both sides of this equation by $g$, the $g^{-1}$ and $g$ cancel out, yielding the equality of the left coset $gH$ and the right coset $Hg$.

Conditions: H is a normal subgroup of G; g is an element of the group G

A normal subgroup satisfies $gH = Hg$ because its defining property is that it is invariant under conjugation, meaning $gHg^{-1} = H$. By right-multiplying both sides of this equation by $g$, the $g^{-1}$ and $g$ cancel out, yielding the equality of the left coset $gH$ and the right coset $Hg$.

Conditions: H is a normal subgroup of G; g is an element of the group G