Let \(\Gamma \) be a graph with vertex set V, and let a, b be nonnegative integers. An (a, b)-regular set in \(\Gamma \) is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in \(V \setminus D\) has exactly b neighbours in D. In particular, a (1, 1)-regular set is called a total perfect code. Let G be a finite group and S a square-free subset of G closed under conjugation. The Cayley sum graph \(\textrm{CayS}(G,S)\) of G is the graph with vertex set G such that two vertices x, y are adjacent if and only if \(xy \in S\). A subset (respectively, subgroup) D of G is called an (a, b)-regular set (respectively, subgroup (a, b)-regular set) of G if there exists a Cayley sum graph of G which admits D as an (a, b)-regular set. We obtain two necessary and sufficient conditions for a subgroup of a finite group G to be a total perfect code in a Cayley sum graph of G. We also obtain two necessary and sufficient conditions for a subgroup of a finite abelian group G to be a total perfect code of G. We classify finite abelian groups whose all non-trivial subgroups of even order are total perfect codes of the group, and as a corollary we obtain that a finite abelian group has the property that every non-trivial subgroup is a total perfect code if and only if it is isomorphic to an elementary abelian 2-group. We prove that, for a subgroup H of a finite abelian group G and any pair of positive integers (a, b) within certain ranges depending on H, H is an (a, b)-regular set of G if and only if it is a total perfect code of G. Finally, we give a classification of subgroup total perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.

令\(\Gamma \)为具有顶点集V的图，并令a、  b为非负整数。\(\Gamma \)中的( a ,  b ) 正则集是V的非空真子集D，使得D中的每个顶点在D中恰好有一个邻居，并且\(V \setminus D\)中的每个顶点恰好有D中的b邻居。特别地，(1, 1)-正则集被称为全完美码。设G为有限群，S为共轭封闭的G的无平方子集。G的凯莱和图\(\textrm{CayS}(G,S)\)是具有顶点集G 的图，使得两个顶点x ,  y相邻当且仅当\(xy \in S\)。如果存在G的凯莱和图 ，其中承认D作为 ​​​​​​​( a ,  b )-常规组。我们得到了有限群G的子群是G的凯莱和图中的全完美码的两个充要条件。我们还得到了有限交换群G的一个子群是G的全完美码的两个充要条件。我们对有限阿贝尔群进行分类，其偶阶的所有非平凡子群都是该群的全完美码，并且作为推论，我们得到有限阿贝尔群具有每个非平凡子群都是全完美码当且仅当的性质如果它与基本阿贝尔 2-群同构。我们证明，对于有限交换群G的子群H以及依赖于H的一定范围内的任何正整数对 ( a ,  b ) ，H是 G 的 ( a , b  )正则集当且仅当是G的全完美码。最后给出了循环群、二面体群和广义四元数群的子群全完美码的分类。