关于同痕群的双扭群环同构问题

中文摘要 AI 辅助

本文考虑经典群环同构问题的一个一般版本，称为双扭群环同构问题（TGRIP），该问题判定有限群的双扭复群代数之间是否存在同构。尽管对于同痕群 $G$ 和 $H$，它们的复群代数是同构的，即 $\mathbb C G \cong \mathbb C H$，但它们的双扭复群代数未必同构。延续这一研究方向，我们建立了两个同痕群具有同构双扭复群代数的充分条件，从而为同痕群提供了（TGRIP）的解答。作为应用，我们研究了秩为 $2$ 的特殊 $p$-群、单中心群以及具有初等阿贝尔 Schur 乘子的类 $2$ 幂零群（均在同痕意义下考虑）的（TGRIP）。我们还给出了几个说明主要结果的例子，包括对阶为 $p^6(p \ge 3)$ 的秩 $2$ 特殊 $p$-群的（TGRIP）的完整解答。

英文摘要

In this article, we consider a general version of the classical group ring isomorphism problem, called the twisted group ring isomorphism problem (TGRIP), which determines an isomorphism between the twisted complex group algebras of finite groups. Although for isoclinic groups $G$ and $H$ their complex group algebras are isomorphic, i.e, $\mathbb C G \cong \mathbb C H$, their twisted complex group algebras need not be isomorphic. Continuing this line of investigation, we establish sufficient conditions under which two isoclinic groups have isomorphic twisted complex group algebras, thereby providing solutions to (TGRIP) for isoclinic groups. As applications, we study (TGRIP) for special $p$-groups of rank $2$, unicentral groups, and nilpotent groups of class $2$ with elementary abelian Schur multipliers, all considered up to isoclinism. We also present several examples illustrating the main results, including a complete solution of (TGRIP) for special $p$-groups of rank $2$ of order $p^6(p \ge 3)$

发表机构

• School of Mathematical Sciences, National Institute of Science Education and Research（国家科学教育研究中心数学科学学院）
• Homi Bhabha National Institute（霍米·巴巴国立研究所）

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