Let \(G_{2}\) be a group which acts trivially on an abelian group \(G_{1}\). According to the Schreier’s Theorem, each 2-cocycle \(\varepsilon \in Z^{2}(G_{2},G_{1})\) determines a group \(G_{\varepsilon }\) which is a central extension of \(G_{1}\) by \(G_{2}\), and we will denote this group by \(G_{1}\underset{\varepsilon }{\times }G_{2} \) and call it the perturbed direct product of \(G_{1}\) by \(G_{2}\) under \(\varepsilon \). The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles \(\varepsilon _{1}\) and \(\varepsilon _{2}\), we find necessary and sufficient conditions for \(G_{1}\underset{\varepsilon _{1}}{\times }G_{2} \) to be isomorphic to \(G_{1}\underset{\varepsilon _{2}}{\times }G_{2}\). Furthermore, we obtain some results about decompositions for a given perturbed direct product \(G_{1}\underset{ \varepsilon }{\times }G_{2}\) when \(G_{1}\) or \(G_{2}\) is a nontrivial direct product.

假设\（G_ {2} \）是对阿贝尔群\（G_ {1} \）起作用的一个组。根据施赖埃尔定理，每个2上闭链\（\ varepsilon \沿Z ^ {2}（G_ {2}，G_ {1}）\）确定一组\（G _ {\ varepsilon} \） ，其是中心的扩展\（G_ {1} \）由\（G_ {2} \），我们将表示该组由\（G_ {1} \底流{\ varepsilon} {\倍} G_ {2} \）和称其为\（\ varepsilon \）下\（G_ {1} \）乘\（G_ {2} \）的摄动直接积。本文的目的是研究被摄动的直接产品的性质。对于两个不同的2-cocycles \（\ varepsilon _ {1} \）和\（\ varepsilon _ {2} \），我们发现\（G_ {1} \ underset {\ varepsilon _ {1}} {\ times} G_ {2} \）同构为\的必要和充分条件（G_ {1} \ underset {\ varepsilon _ {2}} {\ times} G_ {2} \）。此外，对于给定的摄动直接积\（G_ {1} \ underset {\ varepsilon} {\ times} G_ {2} \），当\（G_ {1} \）或\（G_ { 2} \）是非平凡的直接产品。