Are there any differences between $\textrm{Aut}(G)$ and $\textrm{Sym}(G)$ for a group $G$, or are the two notations synonymous?

Question

I have seen both notations--- $\textrm{Aut}(G)$ and $\textrm{Sym}(G)$---for a group $G$, and they seem to denote the same thing: the group of all bijections on $G$.

Answer 1

Every automorphism is a bijection; but the converse is not true. Automorphisms need to be homomorphisms in addition to being bijections. For example, if a bijection does not take the identity element to itself, it can't be an automorphism.

Answer 2

If I had to guess, $\text{Aut}(G)$ is the automorphism group of $G$, while $\text{Sym}(G)$ is the symmetric group on the underlying set of $G$ (a.k.a. the group of all bijections $G \to G$). If $G$ is any nontrivial group, $\text{Aut}(G)$ is a proper subgroup of $\text{Sym}(G)$. You should really look up the conventions in whatever you're reading to make sure they agree with mine.

Answer 3

Consider the maps $l_a: x\mapsto ax$ and $c_a: x\mapsto a^{-1}xa$, where $a$ is a given, non-identity element of $G$. Both are bijections on $G$, but $c_a$ has in addition the operation-preserving property, $c_a(xy)=c_a(x)c_a(y)$ for every $x,y\in G$, which $l_a$ hasn't. So, both are elements of $\operatorname{Sym}(G)$, but $c_a$ only is an element of $\operatorname{Aut}(G)$.