TITLE: Let $A$ be a symmetric subset of a group $G$ such that $A$ contains the identity, and $A$ is covered by some translation. Is then $A$ a subgroup?
QUESTION [5 upvotes]: Let $G$ be a multiplicative group and $A\subseteq G$ such that
1) $\forall a\in A, a^{-1}\in A$
2) $1\in A$
3) $AA \subseteq gA$ for some $g\in G$ 
Can we say that $A$ is a subgroup?

One can immediately show that $g\in A$ and that we have $g^{-1}A \subseteq A \subseteq gA$ but I believe you must use the symmetry property of $A$ (1) again to conclude that it is a subgroup (if it is).

REPLY [8 votes]: $1=1\cdot 1\in AA\subset gA$ so $1=ga_0$ for some $a_0\in A$. Since $a_0=g^{-1}$, we have $g^{-1}\in A$ hence by 1) $g\in A$. We have $gA=A$. Indeed, $A=1\cdot A\subset A\cdot A\subset gA$ and $g^{-1}A\subset gA$ so $A\subset g^2A$. In particular, $1=g^2a$ so $(g^2)^{-1}\in A$ and $g^2\in A$. So $g^2A\subset A\cdot A\subset gA$ and if $x\in gA$ then $x=ga$ for some $a\in A$ so $gx=g^2a\in g^2A\subset gA$ hence $gx=ga'$ for some $a'\in A$ hence $x\in A$ and $gA=A$.  
So $A\cdot A\subset gA=A$, and with 1) and 2) it shows that $A$ is a subgroup of $G$.