symmetric group acting on torus

Question

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ (s_1,\cdots,s_k)\sigma=(s_{\sigma(1)},\cdots,s_{\sigma(k)}). $$ Can $S_k$ be regarded as a subgroup of $T^k$ or not?

Answer 1

It doesn't make sense to conflate automorphisms of an abelian group with elements of the abelian group. Since many notations are ubiquitous in math, such as $\pm1$, you may incidentally find a single symbol used for both an element of a group and an automorphism of a group. This is not the same as saying the automorphism is being viewed as an element of the group, anymore than arbitrarily declaring you will refer to the number two as "apple" and also use the word "apple" to refer to the identity permutation means that the number two is being thought of as the identity map.

We certainly can't view $S_k$ as a subgroup of $\Bbb T^k$; the first is nonabelian, the second isn't.

You can however put $\Bbb T^k$ and $S_k$ in the same group in which conjugating a toral vector by a permutation achieves the effect of having the permutation permute the coordinates of the toral vector. This construction is known as the wreath product $\Bbb T\wr S_k$. As Jyrki mentions in the comments, it is a special case of a semidirect product $K\rtimes H$, in which conjugating elements of $K$ by elements of $H$ achieves the effect of having $H$ act on $K$ by automorphisms.