中文摘要：

组 G 的正常霍尔亚群 N 是有它有它的索引的顺序 coprime 的正常亚群。Schur-Zassenhaus 定理声明每正常霍尔亚群有补充亚群，那是也形成 G 的亚群的一套 coset 代表 H。在这份报纸，我们在场与至少一正常霍尔亚群测试组的同晶型的一个框架，当组们是被给时乘法表。建立框架，我们首先观察到 Schur-Zassenhaus 定理的一个证明是建设性的，并且提出为以正常部分和补充部分的 semidirect 产品，和同晶型的联系行动的严峻的同晶型的一个必要、足够的条件。当正常亚群是 abelian 时，我们然后集中于盒子。由 Le 胆量(STACS 2009 ) 利用有限的组和一种技术的表示理论的基本事实，我们首先让有效同晶型当补充围住发电机的数字时，测试算法。当补充亚群是基本的时，为诉讼， abelian，它未必做围住发电机的数字，我们获得由归结为概括法律同晶型问题测试算法的多项式时间同晶型，它问二线性 subspaces 是否是直到坐标的排列的一样。后者的一个答案能被温和扩展获得单身地指数(在坐标的数字) 为代码同晶型问题的时间算法最近由 Babai 等发展了。(苏打 2011 ) 。Enroute 到获得上述减小，我们在有限的组的表示理论学习下列计算问题：给二个代表并且在 \mathbbZpd \mathbb 上的组 H { Z }_p ^ d， p 一个素数，决定是否在那里存在自守：HH 劝诱的表示 = 并且是相等的，及时 poly (|H| ， p d ) 。

英文摘要：

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur- Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall （STACS 2009）, we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential （in the number of coordinates） time algorithm for code isomorphism problem developed recently by Babai et al. （SODA 2011）. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups： given two representations ρandτ- of a group H over Zp^d, a prime, determine if there exists an automorphism ： ФH→ H, such that the induced representation p Ф= ρ o Ф and τ are equivalent, in time poly（｜H｜, p^d）.