# Laws of Exponents for Cosets or Torsion Subgroups?

I'm trying to understand Arturo Magidin's answer here ― http://math.stackexchange.com/a/120349.

I understand this ― There exists a smallest integer $n$ such that $[T(G)g]^n = T(G)$.

But then he writes ― $\boxed{[T(G)g]^n = T(G)g^n}$. Can someone please explain this?

I've spent an hour looking through my notes, the Internet, and here for Properties or Laws of Exponents for Cosets or Torsion Subgroups. But I can't find anything to justify the boxed ?identity? ? Thanks.

Also, is this true in general for cosets: $\boxed{\text{If H is any subgroup of any group G, then}[Hg]^n = Hg^n}$

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Let me answer your second question first. That statement is false. Take the free product on generators $a, b$. Let $A$ be the subgroup generated by $a$. Then $(Ab)^2 \neq A^2b^2$.
Make sure you read the original question: the group $G$ is assumed to be abelian. It is true that $(Hg)^n = Hg^n$ for an abelian group $G$ with subgroup $H$ and $g \in G$. To see this, note that $(Hg)^n$ is the set of elements of form $h_1gh_2g\cdots h_k g$ with $h_i \in H$ for all $i \in \{1, k\}$. As $G$ is abelian, we can rearrange this to $h_1h_2\cdots h_kg^n$, and $h_1h_2\cdots h_k \in H$ because $H$ is a subgroup.
Thank you Adam Saltz. So $H \text{subgroup of}$ Abelian $G and$g \in G\$ is a sufficient condition. What's the necessary and sufficient condition though for this "exponent rule"? –  Frank Berger Dec 23 '12 at 21:57