Linked Questions

3
votes
3answers
2k views

If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ . [duplicate]

If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ . With the assumption, I dont know how to start the proof. If there is no non-trivial automorphism,...
8
votes
1answer
365 views

Finding all groups with $\text{Aut}(G)=\{1\}$ [duplicate]

Possible Duplicate: $|G|>2$ implies $G$ has non trivial automorphism I am doing this exercise: Find all groups $G$, with $\text{Aut}(G)=\{1\}$. What has been clear to me is the group $G$ ...
2
votes
1answer
2k views

Non trivial Automorphism [duplicate]

Prove that every finite group having more than two elements has a nontrivial Automorphism. It is from Topics in Algebra by Herstein. I am not able to solve.
4
votes
1answer
2k views

Every finite group of order more than two has a nontrivial automorphism [duplicate]

I want to prove that every finite group $G$ of order more than 2 has a nontrivial automorphism. I've seen this question answered on this site for infinite groups, but the proofs given use the fact ...
0
votes
2answers
233 views

Particular nontrivial group has a nonidentity automorphism [duplicate]

If $G$ is a nontrivial group that is not cyclic of order 2, then $G$ has a nonidentity automorphism. This is the exercise of hungerford algebra in the chapter $IV$ MODULES. Can you help me please?
0
votes
1answer
560 views

Prove that if Aut($G$) is the trivial group, then so is $G$? [duplicate]

Let $G$ be a finitely generated group. Show that if Aut($G$) is the trivial group, then so is $G$. I know that if Aut($G$) is the trivial group then $G$ must be abelian but I'm not sure how to use ...
2
votes
1answer
150 views

When the automorphism group is trivial… [duplicate]

If $G$ is a trivial group , obviously $Aut(G)$ is trivial . Does the converse hold $?$ . When we are given that a group $G$ has trivial automorphism group , can we conclude that the ...
0
votes
1answer
163 views

Let $G$ be a group such that $|G| \ge 3$ then $|AutG| \ge 2$. [duplicate]

Let $G$ be a group such that $|G| \ge 3$ then $|AutG| \ge 2$. How can I approach to this problem? It is necessary to divide in cases? For G finite and infinite, or Abelian and non-Abelian? The ...
0
votes
1answer
98 views

Existence of nontrivial automorphism for a group $G$ with $o(G)>2$ [duplicate]

Prove that every finite group having more than two elements has a nontrivial automorphism. Proof: Let $G$ is a group such that $o(G)>2$. Let's consider three following cases: If $G$ is not ...
-1
votes
1answer
43 views

Group, which doesn't have $\mathrm{Aut}(G)$. [duplicate]

Does there some groups which doesn't contain $(\mathrm{Aut}(G),*)$ (only $\mathrm{id}$), where $*$ is composition. I thought that $\mathbb{R}$ could be such group, but I can build bijection between $\...
0
votes
0answers
72 views

every finite group of order greater than 2 has a non-trivial automorphism [duplicate]

Prove that every finite group of order greater than 2 has a non-trivial automorphism. the hint given is that If all of these are trivial, what does this tell us about the group?
0
votes
0answers
46 views

Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
0
votes
1answer
41 views

Must there exist a nontrivial automorphism $\pi$ in $\text{Aut}(G)$? [duplicate]

I was trying to show that there exist a nontrivial automorphism $\pi$ in $\text{Aut}(G)$, and I am taking the case that $G$ is abelian because if $G$ is nonabelian it is trivial. If there exists an $\...
1
vote
0answers
41 views

An infinite group which has only one automorphism [duplicate]

I have known that the only finite group which has only one automorphism is the cyclic group with order less than $2$. But what's the infinite situation? Is there any infinite group which also ...
1
vote
0answers
40 views

groups $G$ such that $Aut(G)$ is trivial [duplicate]

Is it true that, if $G$ is a group such that $Aut(G)$ is trivial, then $G$ is of order at most $2$? Thanks in advance.

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