MHB Can a Group Have a Trivial Automorphism Group with Less than Three Elements?

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Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.
 
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After considerable help from http://math.stackexchange.com/users/98602/mike-miller of MSE (with whom I use to have frequent conversations), I finally solved it.

One impliction is patently obvious, as if $|G| < 3$, then $|G|$ is either $1$, in which case it is straightforward that automorphism of trivial groups is trivial, or $2$ in which case $G \cong \Bbb Z/2\Bbb Z$ and the only automorphism of this group is $x \mapsto x$. For the other implication, we divide the proof into two cases, in which $G$ is abelian and where $G$ is nonabelian :

Claim : $Aut(G)$ is nontrivial for $|G| \geq 3$, given $G$ is abelian.

Proof : As $|G|$ is at least $3$, it contains $1$, $x$ and the inverse $x^{-1}$ of $x$. The map $x \mapsto x^{-1}$ is an automorphism and it's trivial if and only if $x^2 = 1$ for all $x \in G$. Assume it is not the case, then $x \mapsto x^{-1}$ is the desired nontrivial auto of $G$.

If, on the other hand, every element of $G$ has order $2$, then $G$ is a vector space over $\Bbb Z/2\Bbb Z$, i.e., $G = \bigoplus \Bbb Z/2\Bbb Z$. Consider the map $(x_1, x_2, x_3, \cdots) \mapsto (x_2, x_1, x_3, \cdots)$. This is always possible for finite $G$, as there are at least two copies of $\Bbb Z/2\Bbb Z$ sitting inside for all $|G| > 2$. However, if $G$ is infinite, picking two elements of $\Bbb Z/2\Bbb Z$ from infinitely many of the elements requires Zorn's lemma (i.e., Axiom of Choice in disguise). So we have proved that if $G$ is abelian, the statement of the problem holds, conditional on AC.

Claim : $Aut(G)$ is nontrivial for $|G| \geq 3$, given $G$ is nonabelian.

Proof : Consider the conjugation map $x \mapsto axa^{-1}$. This is clearly an automorphism, and it is trivial if and only if $axa^{-1} = x$ which happens if $a$ commutes with $x$. But as $Z(G)$ is nontrivial by assumption, there exists a choice of $a$ such that $ax \neq xa$ for at least one $x \in G$, which proves the existence of a nontrivial map $x \mapsto axa^{-1}$.

This concludes the proof $\blacksquare$
 
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Hi mathbalarka,

There is incoherence in your proof that needs revision.

mathbalarka said:
First, assume that $G$ is abelian. As $|G|$ is at least $3$, it contains $1$, $x$ and the inverse $x^{-1}$ of $x$.The map $x \mapsto x^{-1}$ is an automorphism and it's trivial if and only if $x^2 = 1$ for all $x \in G$. Assume it is not the case, then $x \mapsto x^{-1}$ is the desired nontrivial auto of $G$.

It does not follow from the abelian property of $G$ that $|G| \ge 3$. If you're giving a proof by contradiction, then make that clear from the beginning. Since $x$ is a particular point of $G$, the assignment $x \mapsto x^{-1}$ is not defined as a function from $G$ into $G$.

mathbarlarka said:
If, on the other hand, every element of $G$ has order $2$, then $G$ is a vector space over $\Bbb Z/2\Bbb Z$, i.e., $G = \bigoplus \Bbb Z/2\Bbb Z$. Consider the map swapping two of the $\Bbb Z/2\Bbb Z$s sitting inside, which is an automorphism. This is always possible for finite $G$, as there are at least two copies of $\Bbb Z/2\Bbb Z$ sitting inside for all $|G| > 2$. However, if $G$ is infinite, picking two copies of $\Bbb Z/2\Bbb Z$ from infinitely many of the copies requires Zorn's lemma (i.e., Axiom of Choice in disguise). So we have proved that if $G$ is abelian, the statement of the problem holds, conditional on AC.

You have good ideas, but this part is vague and not carefully written. You can't swap copies in $G$, but elements in $G$.

The first and second paragraphs are not unified. Since the original statement is biconditional you also need to prove the other direction.
 
It does not follow from the abelian property of G that |G|≥3.

You didn't quite get my point. What I have shown is that $Aut(G)$ is nontrivial for $|G| \geq 3$ assuming $G$ is abelian.

Since x is a particular point of G, the assignment x↦x−1 is not defined as a function from G into G.

Huh? The map $G \to G$ as $x \mapsto x^{-1}$ is always defined on $G$. Every element of a group has a unique inverse.

You can't swap copies in G, but elements in G.

I am aware of what I am doing. I believe it is standard terminology to say that $(a, b) \mapsto (b, a)$ swaps the copies of $A$ and $B$.

Seriously, if that's proof writing and very formal terminologies you need instead of the mathematics, I am not going to try writing them out.

EDIT I have edited my post above to make stuffs more clear.
 
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Your proof looks much better now. Thanks for participating, mathbalarka!
 
Suppose $G$ has less than three elements. Then either $G$ is trivial or isomorphic to $\Bbb Z/2\Bbb Z$. In both cases, $\text{Aut}(G)$ is trivial.

The converse will be proven by contradiction. Suppose $G$ has at least three elements. Since $\text{Inn}(G) \le \text{Aut}(G)$ and $\text{Aut}(G)$ is trivial, so is $\text{Inn}(G)$. Therefore, $G$ is abelian. The inversion map is therefore an element of $\text{Aut}(G)$, whence $G$ has exponent two. So $G$ is an $\Bbb F_2$-vector space. Let $e$ be the identity element of $G$. Take distinct elements $g, h \in G\setminus\{e\}$. Using the axiom of choice, choose a basis $\mathcal{B}$ for $G$ that contains $g$ and $h$. Let $T : \mathcal{B} \to \mathcal{B}$ be function such that $T(g) = h$, $T(h) = g$, and $T(x) = x$ for all $x \in \mathcal{B} \setminus\{g,h\}$. Then $T$ extends to a linear isomorphism $\tilde{T}: G \to G$, which is a nontrivial element of $\text{Aut}(G)$. This is a contradiction.
 
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