Let $G$ be a nontrivial group, denote $\text{Aut}(G)$ the group of all its automorphisms of $G$ and denote $S(G)$ the symmetric group on $G$, e.g. the set of all bijections $f:G\rightarrow G$. I would be interested, whether $\text{Aut}(G)$ and $S(G)$ can ever be isomorphic as a groups.
For a finite case, we can make the observation, that any permutation $f\in S(G)$ not fixing the identity cannot be automorphism, so in the finite case $S(G)$ and $\text{Aut}(G)$ aren't even equinumerous.
Could someone provide a rigorous argument for the infinite case? It seems like it is in fact impossible to have these two isomorphic as a groups, but can they atleast have the same cardinality?
We know that if $|G|=\kappa$ then $|S(G)|=2^\kappa$.
E: Adding some more ideas, someone could use:
Denote $S_{fix}(G)$ the set of permutations of $G$ that fix the identity element $e\in G$. Certainly we get
$$
\text{Aut}(G)\preceq S_{fix}(G)\preceq S(G)
$$
If we can show that $S_{fix}(G)$ and $S(G)$ aren't equinumerous, then we are done proving that cardinalities cannot ever be equal.
So, the cardinality problem is solved, now can we show that $\text{Aut}(G)$ and $S(G)$ cannot be isomorphic in the infinite case (or construct a counterexample?).
Best Answer
Suppose $C_\infty^\infty = \langle a_1 \rangle_\infty \times \langle a_2 \rangle_\infty \times \langle a_3 \rangle_\infty \times ...$ is the direct product of countably many isomorphic copies of an infinite cyclic groups. Then $|C_\infty^\infty| = \aleph_0$. And we have $S(\{a_1, a_2, a_3, ...\}) \leq Aut(C_\infty^\infty) \leq S(C_\infty^\infty)$. Thus $2^{\aleph_0} = |S(\{a_1, a_2, a_3, ...\})| \leq |Aut(C_\infty^\infty)| \leq |S(C_\infty^\infty)| = 2^{\aleph_0}$, which results in $|Aut(C_\infty^\infty)| = |S(C_\infty^\infty)|$