In this paper, we exhibit $\textsf{AC}^{3}$ isomorphism tests for coprime extensions $H \ltimes N$ where $H$ is elementary Abelian and $N$ is Abelian; and groups where $\text{Rad}(G) = Z(G)$ is elementary Abelian and $G = \text{Soc}^{*}(G)$. The fact that isomorphism testing for these families is in $\textsf{P}$ was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017). The polynomial-time isomorphism tests for both of these families crucially leveraged small (size $O(\log |G|)$) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that $G$ is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in $\textsf{AC}^{3}$. As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using $\textsf{AC}$ circuits of depth $O(\log^3 n)$ and size $n^{O(\log \log n)}$. This improves upon the previous bound of $n^{O(\log \log n)}$-time due to Grochow and Qiao (ibid.).

翻译：本文针对两类群展示了 $\textsf{AC}^{3}$ 同构测试：一类是余扩张 $H \ltimes N$，其中 $H$ 是初等阿贝尔群，$N$ 是阿贝尔群；另一类是满足 $\text{Rad}(G) = Z(G)$ 为初等阿贝尔群且 $G = \text{Soc}^{*}(G)$ 的群。这些族群的同构测试属于 $\textsf{P}$ 这一事实分别由 Qiao、Sarma 和 Tang（STACS 2011）以及 Grochow 和 Qiao（CCC 2014, SIAM J. Comput. 2017）建立。这两个族群的多项式时间同构测试关键依赖于较小规模（大小为 $O(\log |G|)$）的线性代码等价实例（Babai, SODA 2011）。本文中，我们将 Luks 用于图同构的群论方法（FOCS 1980, J. Comput. Syst. Sci. 1982）与群 $G$ 通过乘法表给出这一事实相结合，在 $\textsf{AC}^{3}$ 中实现了相应的线性代码等价实例。作为本工作的副产品，我们证明了任意中心-根基群（central-radical groups）的同构测试可通过深度为 $O(\log^3 n)$、大小为 $n^{O(\log \log n)}$ 的 $\textsf{AC}$ 电路判定。这改进了 Grochow 和 Qiao（同上）此前 $n^{O(\log \log n)}$ 时间复杂度的界。