Can groups of automorphism over non-isomorphic groups be isomorphic?

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Can two groups of automorphisms over non-isomorphic groups be isomorphic?
If $G$ and $G'$ are non-isomorphic groups and $textAut(G)$ and $textAut(G')$ be their group of automorphisms, then can they be isomorphic to each other or not?

group-theory automorphism-group

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edited Jul 29 at 7:52

Taroccoesbrocco

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asked Jul 29 at 7:34

ke shradha

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up vote
0
down vote

favorite

Can two groups of automorphisms over non-isomorphic groups be isomorphic?
If $G$ and $G'$ are non-isomorphic groups and $textAut(G)$ and $textAut(G')$ be their group of automorphisms, then can they be isomorphic to each other or not?

group-theory automorphism-group

share|cite|improve this question

edited Jul 29 at 7:52

Taroccoesbrocco

3,31941331

asked Jul 29 at 7:34

ke shradha

141

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Can two groups of automorphisms over non-isomorphic groups be isomorphic?
If $G$ and $G'$ are non-isomorphic groups and $textAut(G)$ and $textAut(G')$ be their group of automorphisms, then can they be isomorphic to each other or not?

group-theory automorphism-group

share|cite|improve this question

edited Jul 29 at 7:52

Taroccoesbrocco

3,31941331

asked Jul 29 at 7:34

ke shradha

141

Can two groups of automorphisms over non-isomorphic groups be isomorphic?
If $G$ and $G'$ are non-isomorphic groups and $textAut(G)$ and $textAut(G')$ be their group of automorphisms, then can they be isomorphic to each other or not?

group-theory automorphism-group

share|cite|improve this question

edited Jul 29 at 7:52

Taroccoesbrocco

3,31941331

asked Jul 29 at 7:34

ke shradha

141

share|cite|improve this question

share|cite|improve this question

edited Jul 29 at 7:52

Taroccoesbrocco

3,31941331

edited Jul 29 at 7:52

Taroccoesbrocco

3,31941331

edited Jul 29 at 7:52

Taroccoesbrocco

3,31941331

3,31941331

asked Jul 29 at 7:34

ke shradha

141

asked Jul 29 at 7:34

ke shradha

141

asked Jul 29 at 7:34

ke shradha

141

141

add a comment | 

add a comment | 

2 Answers
2

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up vote
3
down vote

Aut$(Bbb Z_3)$ and Aut$(Bbb Z_4)$ are both isomorphic to $Bbb Z_2$.

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answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
  – Oiler
  Jul 29 at 8:05
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And What is their automorphism?
  – ke shradha
  Jul 29 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @keshradha the trivial one and inversion.
  – Henrique Augusto Souza
  Jul 29 at 15:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is inversion?
  – ke shradha
  Jul 30 at 7:31
  

  
  
  
  

add a comment | 

up vote
0
down vote

A non-abelian example, $Aut(Q) cong S_4 cong Aut(S_4)$. Here $Q$ is the quaternion group of $8$ elements.

share|cite|improve this answer

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are they always same...can u give an example where they are not same but isomorphic
  – ke shradha
  Jul 30 at 7:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
  – Nicky Hekster
  Jul 30 at 8:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
  – ke shradha
  Jul 30 at 10:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
  – Nicky Hekster
  Jul 30 at 20:54
  

  
  
  
  

add a comment | 

Your Answer

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote

Aut$(Bbb Z_3)$ and Aut$(Bbb Z_4)$ are both isomorphic to $Bbb Z_2$.

share|cite|improve this answer

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
  – Oiler
  Jul 29 at 8:05
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And What is their automorphism?
  – ke shradha
  Jul 29 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @keshradha the trivial one and inversion.
  – Henrique Augusto Souza
  Jul 29 at 15:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is inversion?
  – ke shradha
  Jul 30 at 7:31
  

  
  
  
  

add a comment | 

up vote
3
down vote

Aut$(Bbb Z_3)$ and Aut$(Bbb Z_4)$ are both isomorphic to $Bbb Z_2$.

share|cite|improve this answer

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
  – Oiler
  Jul 29 at 8:05
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And What is their automorphism?
  – ke shradha
  Jul 29 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @keshradha the trivial one and inversion.
  – Henrique Augusto Souza
  Jul 29 at 15:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is inversion?
  – ke shradha
  Jul 30 at 7:31
  

  
  
  
  

add a comment | 

up vote
3
down vote

up vote
3
down vote

Aut$(Bbb Z_3)$ and Aut$(Bbb Z_4)$ are both isomorphic to $Bbb Z_2$.

share|cite|improve this answer

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

Aut$(Bbb Z_3)$ and Aut$(Bbb Z_4)$ are both isomorphic to $Bbb Z_2$.

share|cite|improve this answer

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

share|cite|improve this answer

share|cite|improve this answer

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

answered Jul 29 at 7:36

Lord Shark the Unknown

84.5k950111

84.5k950111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
  – Oiler
  Jul 29 at 8:05
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And What is their automorphism?
  – ke shradha
  Jul 29 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @keshradha the trivial one and inversion.
  – Henrique Augusto Souza
  Jul 29 at 15:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is inversion?
  – ke shradha
  Jul 30 at 7:31
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
  – Oiler
  Jul 29 at 8:05
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And What is their automorphism?
  – ke shradha
  Jul 29 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @keshradha the trivial one and inversion.
  – Henrique Augusto Souza
  Jul 29 at 15:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What is inversion?
  – ke shradha
  Jul 30 at 7:31
  

  
  
  
  

An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
– Oiler
Jul 29 at 8:05

An infinite example of such groups: $operatornameAut(mathbbZ/p mathbbZ)$ and $operatornameAut( mathbbZ/2p mathbbZ)$ for prime $p$.
– Oiler
Jul 29 at 8:05

And What is their automorphism?
– ke shradha
Jul 29 at 10:39

And What is their automorphism?
– ke shradha
Jul 29 at 10:39

@keshradha the trivial one and inversion.
– Henrique Augusto Souza
Jul 29 at 15:42

@keshradha the trivial one and inversion.
– Henrique Augusto Souza
Jul 29 at 15:42

What is inversion?
– ke shradha
Jul 30 at 7:31

What is inversion?
– ke shradha
Jul 30 at 7:31

add a comment | 

up vote
0
down vote

A non-abelian example, $Aut(Q) cong S_4 cong Aut(S_4)$. Here $Q$ is the quaternion group of $8$ elements.

share|cite|improve this answer

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are they always same...can u give an example where they are not same but isomorphic
  – ke shradha
  Jul 30 at 7:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
  – Nicky Hekster
  Jul 30 at 8:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
  – ke shradha
  Jul 30 at 10:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
  – Nicky Hekster
  Jul 30 at 20:54
  

  
  
  
  

add a comment | 

up vote
0
down vote

A non-abelian example, $Aut(Q) cong S_4 cong Aut(S_4)$. Here $Q$ is the quaternion group of $8$ elements.

share|cite|improve this answer

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are they always same...can u give an example where they are not same but isomorphic
  – ke shradha
  Jul 30 at 7:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
  – Nicky Hekster
  Jul 30 at 8:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
  – ke shradha
  Jul 30 at 10:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
  – Nicky Hekster
  Jul 30 at 20:54
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

A non-abelian example, $Aut(Q) cong S_4 cong Aut(S_4)$. Here $Q$ is the quaternion group of $8$ elements.

share|cite|improve this answer

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

A non-abelian example, $Aut(Q) cong S_4 cong Aut(S_4)$. Here $Q$ is the quaternion group of $8$ elements.

share|cite|improve this answer

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

share|cite|improve this answer

share|cite|improve this answer

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

answered Jul 29 at 14:32

Nicky Hekster

26.9k53052

26.9k53052

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are they always same...can u give an example where they are not same but isomorphic
  – ke shradha
  Jul 30 at 7:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
  – Nicky Hekster
  Jul 30 at 8:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
  – ke shradha
  Jul 30 at 10:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
  – Nicky Hekster
  Jul 30 at 20:54
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are they always same...can u give an example where they are not same but isomorphic
  – ke shradha
  Jul 30 at 7:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
  – Nicky Hekster
  Jul 30 at 8:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
  – ke shradha
  Jul 30 at 10:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
  – Nicky Hekster
  Jul 30 at 20:54
  

  
  
  
  

Are they always same...can u give an example where they are not same but isomorphic
– ke shradha
Jul 30 at 7:32

Are they always same...can u give an example where they are not same but isomorphic
– ke shradha
Jul 30 at 7:32

What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
– Nicky Hekster
Jul 30 at 8:59

What do you exactly mean? Two isomorphic groups with non-isomorphic automorphism groups? Those do not exist!
– Nicky Hekster
Jul 30 at 8:59

I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
– ke shradha
Jul 30 at 10:17

I mean two non-isomorphic groups whose automorphism groups are only isomorphic they should not be same...like ur example both are $S_4$
– ke shradha
Jul 30 at 10:17

OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
– Nicky Hekster
Jul 30 at 20:54

OK (also have a look at math.stackexchange.com/questions/586471/…), let $p geq 5$ be prime. Then $Aut(Q times C_p) cong S_4 times C_p-1 cong Aut(S_4 times C_p)$. Now you have three non-isomorphic non-abelian groups.
– Nicky Hekster
Jul 30 at 20:54

add a comment | 

 

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