Mixing
Number of automorphisms on alternating group $A_n$.
0
$begingroup$
How do I find the number of automorphisms of the alternating group on $n$ symbols $A_n$? I have found it for cyclic groups. But I have no idea how to do it here. In particular, I was asked to find it for $n=4 $. Please help me. I'll be thankful if to you if you suggest me how to approach and what I should read again.
group-theory finite-groups automorphism-group
edited Jan 17 at 15:08
the_fox
2,89021537
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
$endgroup$
add a comment |
0
$begingroup$
How do I find the number of automorphisms of the alternating group on $n$ symbols $A_n$? I have found it for cyclic groups. But I have no idea how to do it here. In particular, I was asked to find it for $n=4 $. Please help me. I'll be thankful if to you if you suggest me how to approach and what I should read again.
group-theory finite-groups automorphism-group
edited Jan 17 at 15:08
the_fox
2,89021537
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
$endgroup$
2
$begingroup$
See, for example, this reference
$endgroup$
– lulu
Jan 17 at 12:17
$begingroup$
I'm final year undergraduate student. I don't know the concept of outer automorphisms. And thank you for quick link.
$endgroup$
– Ravi Satpute
Jan 17 at 12:27
1
$begingroup$
You should look that up, it's fundamental to the notion of automorphism groups. The key point is that the Inner automorphisms (those induced by conjugation) form a normal subgroup of $Aut(G)$. The outer automorphism group is the quotient.
$endgroup$
– lulu
Jan 17 at 12:31
1
$begingroup$
Note: the automorphism groups for symmetric and alternating groups are a rather technical topic..though the final result (at least for large enough $n$) is simple enough. (But the exotic automorphism of $S_6$ is worth a lengthy story on its own). I'd think it was much more important to learn basic notions (like inner and outer automorphisms) first.
$endgroup$
– lulu
Jan 17 at 12:33
add a comment |
0
0
0
$begingroup$
How do I find the number of automorphisms of the alternating group on $n$ symbols $A_n$? I have found it for cyclic groups. But I have no idea how to do it here. In particular, I was asked to find it for $n=4 $. Please help me. I'll be thankful if to you if you suggest me how to approach and what I should read again.
group-theory finite-groups automorphism-group
edited Jan 17 at 15:08
the_fox
2,89021537
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
$endgroup$
How do I find the number of automorphisms of the alternating group on $n$ symbols $A_n$? I have found it for cyclic groups. But I have no idea how to do it here. In particular, I was asked to find it for $n=4 $. Please help me. I'll be thankful if to you if you suggest me how to approach and what I should read again.
group-theory finite-groups automorphism-group
group-theory finite-groups automorphism-group
edited Jan 17 at 15:08
the_fox
2,89021537
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
edited Jan 17 at 15:08
the_fox
2,89021537
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
edited Jan 17 at 15:08
the_fox
2,89021537
edited Jan 17 at 15:08
the_fox
2,89021537
edited Jan 17 at 15:08
the_fox
2,89021537
2,89021537
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
asked Jan 17 at 12:13
Ravi SatputeRavi Satpute
267
267
2
$begingroup$
See, for example, this reference
$endgroup$
– lulu
Jan 17 at 12:17
$begingroup$
I'm final year undergraduate student. I don't know the concept of outer automorphisms. And thank you for quick link.
$endgroup$
– Ravi Satpute
Jan 17 at 12:27
1
$begingroup$
You should look that up, it's fundamental to the notion of automorphism groups. The key point is that the Inner automorphisms (those induced by conjugation) form a normal subgroup of $Aut(G)$. The outer automorphism group is the quotient.
$endgroup$
– lulu
Jan 17 at 12:31
1
$begingroup$
Note: the automorphism groups for symmetric and alternating groups are a rather technical topic..though the final result (at least for large enough $n$) is simple enough. (But the exotic automorphism of $S_6$ is worth a lengthy story on its own). I'd think it was much more important to learn basic notions (like inner and outer automorphisms) first.
$endgroup$
– lulu
Jan 17 at 12:33
add a comment |
2
$begingroup$
See, for example, this reference
$endgroup$
– lulu
Jan 17 at 12:17
$begingroup$
I'm final year undergraduate student. I don't know the concept of outer automorphisms. And thank you for quick link.
$endgroup$
– Ravi Satpute
Jan 17 at 12:27
1
$begingroup$
You should look that up, it's fundamental to the notion of automorphism groups. The key point is that the Inner automorphisms (those induced by conjugation) form a normal subgroup of $Aut(G)$. The outer automorphism group is the quotient.
$endgroup$
– lulu
Jan 17 at 12:31
1
$begingroup$
Note: the automorphism groups for symmetric and alternating groups are a rather technical topic..though the final result (at least for large enough $n$) is simple enough. (But the exotic automorphism of $S_6$ is worth a lengthy story on its own). I'd think it was much more important to learn basic notions (like inner and outer automorphisms) first.
$endgroup$
– lulu
Jan 17 at 12:33
2
2
$begingroup$
See, for example, this reference
$endgroup$
– lulu
Jan 17 at 12:17
$begingroup$
See, for example, this reference
$endgroup$
– lulu
Jan 17 at 12:17
$begingroup$
I'm final year undergraduate student. I don't know the concept of outer automorphisms. And thank you for quick link.
$endgroup$
– Ravi Satpute
Jan 17 at 12:27
$begingroup$
I'm final year undergraduate student. I don't know the concept of outer automorphisms. And thank you for quick link.
$endgroup$
– Ravi Satpute
Jan 17 at 12:27
1
1
$begingroup$
You should look that up, it's fundamental to the notion of automorphism groups. The key point is that the Inner automorphisms (those induced by conjugation) form a normal subgroup of $Aut(G)$. The outer automorphism group is the quotient.
$endgroup$
– lulu
Jan 17 at 12:31
$begingroup$
You should look that up, it's fundamental to the notion of automorphism groups. The key point is that the Inner automorphisms (those induced by conjugation) form a normal subgroup of $Aut(G)$. The outer automorphism group is the quotient.
$endgroup$
– lulu
Jan 17 at 12:31
1
1
$begingroup$
Note: the automorphism groups for symmetric and alternating groups are a rather technical topic..though the final result (at least for large enough $n$) is simple enough. (But the exotic automorphism of $S_6$ is worth a lengthy story on its own). I'd think it was much more important to learn basic notions (like inner and outer automorphisms) first.
$endgroup$
– lulu
Jan 17 at 12:33
$begingroup$
Note: the automorphism groups for symmetric and alternating groups are a rather technical topic..though the final result (at least for large enough $n$) is simple enough. (But the exotic automorphism of $S_6$ is worth a lengthy story on its own). I'd think it was much more important to learn basic notions (like inner and outer automorphisms) first.
$endgroup$
– lulu
Jan 17 at 12:33
add a comment |
0
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2
$begingroup$
See, for example, this reference
$endgroup$
– lulu
Jan 17 at 12:17
$begingroup$
I'm final year undergraduate student. I don't know the concept of outer automorphisms. And thank you for quick link.
$endgroup$
– Ravi Satpute
Jan 17 at 12:27
1
$begingroup$
You should look that up, it's fundamental to the notion of automorphism groups. The key point is that the Inner automorphisms (those induced by conjugation) form a normal subgroup of $Aut(G)$. The outer automorphism group is the quotient.
$endgroup$
– lulu
Jan 17 at 12:31
1
$begingroup$
Note: the automorphism groups for symmetric and alternating groups are a rather technical topic..though the final result (at least for large enough $n$) is simple enough. (But the exotic automorphism of $S_6$ is worth a lengthy story on its own). I'd think it was much more important to learn basic notions (like inner and outer automorphisms) first.
$endgroup$
– lulu
Jan 17 at 12:33