In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fra\"iss\'e bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) heirarchy. This is a Spoiler-Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler-Leman (WL) coloring, which we call 2-ary WL. We then show that the 2-ary WL is equivalent to the second Ehrenfeucht-Fra\"iss\'e bijective pebble game in Hella's heirarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (\emph{ibid.}), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $O(1)$ variables and $O(1)$ quantifier depth.

翻译：本文通过研究Hella (Ann. Pure Appl. Log., 1989)层级中第二个Ehrenfeucht-Fraïssé双射卵石博弈的能力，探讨了有限群的描述复杂性理论。这是一个破坏者-复制者博弈，其中破坏者每轮最多可放置两个卵石。尽管该博弈平凡地解决了图同构问题，但对于有限群及其他三元关系结构而言可能并不平凡。我们首先提出Weisfeiler-Leman (WL)着色的一种新型推广，称为二元WL。随后证明二元WL等价于Hella层级中的第二个Ehrenfeucht-Fraïssé双射卵石博弈。我们的主要结论是，在卵石博弈表征中，仅需$O(1)$个卵石和$O(1)$轮即可识别所有无非交换正规子群的群（此类群的同构测试已知属于$\mathsf{P}$；Babai, Codenotti, & Qiao, ICALP 2012）。特别地，我们证明在前几轮内，破坏者可迫使复制者在后续每一轮中选取两个此类群之间的同构。根据Hella的结果（同上），这等价于说这些群可由带广义二元量词的一阶逻辑公式识别，且仅使用$O(1)$个变量和$O(1)$量词深度。