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For instance, the condition $(1)$ for a triple $(\alpha, \beta, \gamma) \in \mathbb{F}_q^3$ holds if at least two of $\alpha, \beta$ and $\gamma$ are zero. The image in $\text{PSL}_2(\mathbb{F}_q)$ of a pair $(A, B) \in \text{SL}_2(\mathbb{F}_q)^2$ such that $\text{Tr}(A, B) = (\alpha, \beta, \gamma)$ generates a dihedral group in this case. Condition $(2)$ holds if $\alpha^2 + \beta^2 + \gamma^2 - \alpha \beta \gamma - 2 = 2$, the left-hand side being the Fricke polynomial. This corresponds to the case of an affine subgroup of $\text{PSL}_2(\mathbb{F}_q)$. (A subgroup of $\text{PSL}_2(\mathbb{F}_q)$ is said to be affine if it is conjugate to a subgroup of the image in $\text{PSL}_2(\mathbb{F}_q)$ of the subgroup of upper triangular matrices in $\text{SL}_2(\mathbb{F}_q)$.) Condition $(4)$ holds if $\alpha, \beta$ and $\gamma$ lie in a proper subfield of $\mathbb{F}_q$. The remaining conditions are presented in the Addendum below.

For instance, the condition $(1)$ for a triple $(\alpha, \beta, \gamma) \in \mathbb{F}_q^3$ holds if at least two of $\alpha, \beta$ and $\gamma$ are zero. The image in $\text{PSL}_2(\mathbb{F}_q)$ of a pair $(A, B) \in \text{SL}_2(\mathbb{F}_q)^2$ such that $\text{Tr}(A, B) = (\alpha, \beta, \gamma)$ generates a dihedral group in this case. Condition $(2)$ holds if $\alpha^2 + \beta^2 + \gamma^2 - \alpha \beta \gamma - 2 = 2$, the left-hand side being the Fricke polynomial. This corresponds to the case of an affine subgroup of $\text{PSL}_2(\mathbb{F}_q)$, that is a subgroup which is conjugate to a subgroup of the image in $\text{PSL}_2(\mathbb{F}_q)$ of the subgroup of upper triangular matrices in $\text{SL}_2(\mathbb{F}_q)$. Condition $(4)$ holds if $\alpha, \beta$ and $\gamma$ lie in a proper subfield of $\mathbb{F}_q$. The remaining conditions are presented in the Addendum below.

Addendum. Let $Q(\alpha, \beta, \gamma) \Doteq \alpha^2 + \beta^2 + \gamma^2 - \alpha \beta \gamma - 2$ be the Fricke polynomial. Here are the five conditions derived by Daryl McCullough and Marcus Wanderley from Dickson's Subgroup Classification Theorem and Macbeath's Fricke polynomial criterion:

Addendum. Let $Q(\alpha, \beta, \gamma) \Doteq \alpha^2 + \beta^2 + \gamma^2 - \alpha \beta \gamma - 2$ be the Fricke polynomial. This name was coined because of Fricke's trace identity $$\text{Tr}([A, B]) = Q(\text{Tr}(A), \text{Tr}(B), \text{Tr}(AB)).$$ Here are the five conditions derived by Daryl McCullough and Marcus Wanderley from Dickson's Subgroup Classification Theorem and Macbeath's Fricke polynomial criterion:

• deciding whether two elements of $\mathbb{F}_q$ are equal.
• deciding whether $n$ elements of $\mathbb{F}_q$ ($n \le 4$) generate a proper subfield.
• deciding whether an element of $\mathbb{F}_q$ belongs to a subfield of $\mathbb{F}_q$ generated by $n$ given elements ($n \le 4$).

• deciding whether two elements of $\mathbb{F}_q$ are equal.
• deciding whether $k$ elements of $\mathbb{F}_q$ ($k \le 4$) generate a proper subfield.
• deciding whether an element of $\mathbb{F}_q$ belongs to a subfield of $\mathbb{F}_q$ generated by $k$ given elements ($k \le 4$).