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Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.

1. Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

2. Jordan: Suppose that $G$ is a finite linear group of degree $n$ over a field of characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most \beta(n)$.

3. Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

4. Platonov: Suppose that $G$ is a linear group of degree $n$ of finite Pruefer rang $r$ over a field of characteristic $p > 0$. Then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

5. Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.

Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.

1. Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

2. Jordan: Suppose that $G$ is a finite linear group of degree $n$ over a field of characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most $\beta(n)$.

3. Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

4. Platonov: Suppose that $G$ is a linear group of degree $n$ of finite Pruefer rang $r$ over a field of characteristic $p > 0$. Then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

5. Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.

Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.

1. Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

2. Jordan: Suppose that $G$ is a finite linear group of degree $n$ over a field characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most $\beta(n)$.

3. Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

4. Platonov: Suppose that $G$ is a linear group of degree $n$ of finite Pruefer rang $r$ over a field of characteristic $p > 0$. Then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

5. Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.

Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.

1. Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

2. Jordan: Suppose that $G$ is a finite linear group of degree $n$ over a field of characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most \beta(n)$.

3. Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

4. Platonov: Suppose that $G$ is a linear group of degree $n$ of finite Pruefer rang $r$ over a field of characteristic $p > 0$. Then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

5. Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.

Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.

1. Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

2. Jordan: Suppose that $G$ is a finite linear group over a field characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most $\beta(n)$.

3. Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

4. Platonov: Suppose that $G$ is a linear group of finite Pruefer rang $r$. Then $G$ is abelian-by-finite, and if the characteristic of the field is $ p > 0$, then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

5. Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.

Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.

1. Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

2. Jordan: Suppose that $G$ is a finite linear group of degree $n$ over a field characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most $\beta(n)$.

3. Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

4. Platonov: Suppose that $G$ is a linear group of degree $n$ of finite Pruefer rang $r$ over a field of characteristic $p > 0$. Then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

5. Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.