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) hierarchy. 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 2-ary WL is equivalent to the second Ehrenfeucht--Fra\"iss\'e bijective pebble game in Hella's hierarchy. 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). We actually show that $7$ pebbles and $7$ rounds suffice. 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 (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

翻译：本文通过考察Hella层次结构(Ann. Pure Appl. Log., 1989)中二阶Ehrenfeucht–Fraïssé双射鹅卵石博弈的判别能力，探索有限群的描述复杂性理论。这是一种Spoiler–Duplicator博弈，其中Spoiler每轮最多可放置两枚鹅卵石。虽然该博弈能平凡地解决图同构问题，但对于有限群及其他三元关系结构可能具有非平凡意义。我们首先提出Weisfeiler–Leman(WL)着色方法的新推广，称为二元WL。随后证明二元WL等价于Hella层次结构中的二阶Ehrenfeucht–Fraïssé双射鹅卵石博弈。我们的主要结果表明：在鹅卵石博弈刻画中，仅需$O(1)$枚鹅卵石和$O(1)$轮即可识别所有无阿贝尔正规子群的群（该类群的同构判定已知属于$\mathsf{P}$；Babai, Codenotti & Qiao, ICALP 2012）。我们实际证明$7$枚鹅卵石和$7$轮即足够。特别地，我们证明在前几轮内，Spoiler可迫使Duplicator在后续每轮中选择两个此类群之间的同构映射。根据Hella的结果(ibid.)，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，且仅需$7$个变量和$7$层量词深度。