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Let the group ring $\mathbb{Z}_2S_3,$ where $S_3$ is the permutation group on $3$ elements, with presentation $S_3 = \{1, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau \} = \langle \sigma, \tau | \sigma^3 = \tau^2 = 1, \tau \sigma = \sigma^2 \tau \rangle $

Let $\mathfrak{X} = \left\{ \sum\limits_{g \in S_3} a_gg \Big| \sum\limits_{g \in S_3} a_g = 0 \right\} = \langle \sigma + \tau \rangle$ (I guess)

My questions are:

How can I compute $\mathfrak{X}^k$, where $k \ge 2?$

There is possible generalize this method to obtain the powers $\mathfrak{X}^k$ of $\mathfrak{X} = \left\{ \sum\limits_{g \in S_m} a_gg \Big| \sum\limits_{g \in S_m} a_g = 0 \right\} \subset \mathbb{Z}_nS_m?$

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