Algorithms for continuous and discrete abelian groups and their related classes

In this paper, algorithms are developed to determine properties of finite Z-modules (i.e. abelian groups) related to the classes of (quasi)continuous and (quasi)discrete groups. To this end, some algorithms are provided to check whether the groups have the summand intersection (sum) property, generalized summand intersection (sum) property or absolute direct summand property, and to check whether the groups are virtually semisimple, virtually C2, virtually extending, C2, C3, D2, D3, (quasi)continuous or (quasi)discrete. The algorithms are implemented and published as a collection of GAP4 functions named <b>codiaga</b> (see Ö. Taşdemir and A. Carus [<i>A collection of gap algorithms related to continuous and discrete abelian groups</i>. Available at: https://codiaga.trakya.edu.tr] ).本文开发了相关算法，用于确定与（拟）连续（quasi-continuous）及（拟）离散（quasi-discrete）交换群类相关的有限Z-模（Z-module，即交换群）的性质。为此，本文提供了多组算法，可用于判定群是否具备直和交（和）性质（summand intersection (sum) property）、广义直和交（和）性质（generalized summand intersection (sum) property）或绝对直和性质（absolute direct summand property），同时可验证群是否为几乎半单（virtually semisimple）、几乎C2（virtually C2）、几乎扩张（virtually extending）、C2、C3、D2、D3、（拟）连续或（拟）离散群。上述算法已被实现并发布为一套名为**codiaga**的GAP4函数集，详见Ö. Taşdemir与A. Carus所著《A collection of gap algorithms related to continuous and discrete abelian groups》，可通过链接https://codiaga.trakya.edu.tr 获取。