Parallel Algorithms for Group Isomorphism via Code Equivalence

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.).