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Nearly Linear Time Isomorphism Algorithms for Some Nonabelian Group Classes
Theory of Computing Systems ( IF 0.5 ) Pub Date : 2020-10-21 , DOI: 10.1007/s00224-020-10010-z
Bireswar Das , Shivdutt Sharma

The isomorphism problem for groups, when the groups are given by their Cayley tables is a well-studied problem. This problem has been studied for various restricted classes of groups. Kavitha gave a linear time isomorphism algorithm for abelian groups (JCSS 2007). Although there are isomorphism algorithms for certain nonabelian group classes represented by their Cayley tables, the complexities of those algorithms are usually super-linear. In this paper, we design linear and nearly linear time isomorphism algorithms for some nonabelian groups. More precisely,

  • We design a linear-time algorithm to factor Hamiltonian groups. This allows us to obtain an \(\mathcal {O}(n)\) algorithm for the isomorphism problem of Hamiltonian groups, where n is the order of the groups.

  • We design a nearly linear time algorithm to find a maximal abelian direct factor of an input group. As a byproduct we obtain an \(\tilde {\mathcal {O}}(n)\) isomorphism for groups that can be decomposed as a direct product of a nonabelian group of bounded order and an abelian group, where n is the order of the groups.

  • We observe that testing normality, computing the center of a group, finding a logarithmic sized generating set, computing quotient groups for groups given by their Cayley table could be done in linear or nearly linear time.



中文翻译:

某些Nonabelian群类的近似线性时间同构算法

当组由其Cayley表给出时,组的同构问题是一个经过充分研究的问题。已经针对各种受限类别的组研究了此问题。Kavitha为阿贝尔群提供了线性时间同构算法(JCSS 2007)。尽管对于某些非阿贝尔群类,它们的Cayley表表示了同构算法,但是这些算法的复杂性通常是超线性的。在本文中,我们为一些非阿贝尔群设计了线性和近似线性时间同构算法。更确切地说,

  • 我们设计了一个线性时间算法来分解哈密顿群。这使我们能够为哈密顿群的同构问题获得\(\ mathcal {O}(n)\)算法,其中n是群的阶。

  • 我们设计了一种近似线性的时间算法,以找到输入组的最大阿贝尔直接因数。作为副产品,我们获得了\(\ tilde {\ mathcal {O}}(n)\)同构,这些同构可以分解为有界阶的非阿贝尔群和阿贝尔群的直接乘积,其中n是阶组中的。

  • 我们观察到测试正态性,计算组的中心,找到对数大小的生成集,计算由其Cayley表给出的组的商组可以在线性或近似线性时间内完成。

更新日期:2020-10-30
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