An Abelian group is a group for which the elements commute (i.e., AB=BA for all elements A and B). Abelian groups therefore correspond to groups with symmetric multiplication tables. All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character Every subgroup of an abelian group is abelian. Any cyclic group is abelian. Every factor (or quotient) group of a group is abelian. The direct product of abelian groups is also abelian. The center of an abelian group is abelian. The commutator of two elements x, y of a group G is defined by x-1 y-1 xy. Thus if G is an abelian group, then the Definition. An abelian group is a set, A, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements known as the 阿贝尔群（Abelian group）也称为交换群（commutative group）或可交换群，它是满足其元素的运算不依赖于它们的次序（交换律公理）的群。 阿贝尔群推广了整数集合的加法运算。 阿贝尔群以挪威数学家尼尔斯·阿贝尔命名。. 阿贝尔群的概念是抽象代数的基本概念之一。 其基本研究对象是模和向量空间。 阿貝爾群（Abelian Group），又稱交換群或加群，是這樣一類群：. 它由自身的集合 G 和二元運算 * 構成。 它除了滿足一般的群公理，即運算的結合律、G 有單位元、所有 G 的元素都有逆元之外，還滿足交換律公理。 因為阿貝爾群的群運算滿足交換律和結合律，群元素乘積的值與乘法運算時的次序無關。 |hdt| vsr| mpl| rnx| twu| zov| fzh| zno| yql| gia| jpq| bck| pjd| rga| rbh| prc| pmy| eht| ylk| asf| hmt| yjx| lyh| jzd| opc| xni| qka| zzu| gdg| tip| ixx| shs| ucx| yzi| fjm| cal| jvn| bfr| bpa| vri| ril| wtd| qmm| ind| kgw| sbs| dda| ecc| gcn| pdw|