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Just turning comments into an answer.

1. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are here.

2. A list of the type you asked for does exist thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.

Just turning comments into an answer.

1. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are here.

2. A list of the type you asked for does exist thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.

Just turning comments into an answer.

1. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are here.

2. A list of the type you asked for does exist thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.

Just turning comments into an answer.

1. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are here.

2. A list of the type you asked for does exist thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.

Just turning comments into an answer.

1. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 5$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are here.

2. A list of the type you asked for does exist thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.

Just turning comments into an answer.

1. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are here.

2. A list of the type you asked for does exist thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.