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This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)

Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

EDIT: In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative." You can read about it in section 1 of this paper of Katz.

For the specific problem of distinguishing a subgroup from a group by means of moments,the 2005 paper of Guralnick and Tiep is relevant.

This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)

Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

EDIT: In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative." You can read about it in section 1 of this paper of Katz.

For the specific problem of distinguishing a subgroup from a group by means of moments,  the 2005 paper of Guralnick and Tiep is relevant.

This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)

Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

EDIT: In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative." You can read about it in section 1 of this paper of Katz.

For the specific problem of distinguishing a subgroup from a group by means of moments, see the 2005 paper of Guralnick and Tiep is relevant.

This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)

Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

EDIT: In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative." You can read about it in section 1 of this paper of Katz.

For the specific problem of distinguishing a subgroup from a group by means of moments,the 2005 paper of Guralnick and Tiep is relevant.

This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)

Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)

Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

EDIT: In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative." You can read about it in section 1 of this paper of Katz.

For the specific problem of distinguishing a subgroup from a group by means of moments, see the 2005 paper of Guralnick and Tiep is relevant.