We use the group Fourier transform over the symmetric group $S_n$ to reverse engineer a 1-layer feedforward network that has "grokked" the multiplication of $S_5$ and $S_6$. Each model discovers the true subgroup structure of the full group and converges on circuits that decompose the group multiplication into the multiplication of the group's conjugate subgroups. We demonstrate the value of using the symmetries of the data and models to understand their mechanisms and hold up the ``coset circuit'' that the model uses as a fascinating example of the way neural networks implement computations. We also draw attention to current challenges in conducting mechanistic interpretability research by comparing our work to Chughtai et al. [6] which alleges to find a different algorithm for this same problem.

使用对称群$S_n$上的群Fourier变换，逆向工程出一个能够理解$S_5$和$S_6$乘法的一层前馈网络。每个模型都发现了完整群的真实子群结构，并收敛于将群乘法分解为群的共轭子群乘法的电路。我们展示了利用数据和模型的对称性来理解它们的机制的价值，并举出了模型使用的“余类电路”作为神经网络实现计算的迷人示例。同时，通过将我们的工作与Chughtai等人的工作进行比较，我们还关注了当前进行机制可解释性研究的挑战。

掌握陪集的群乘法