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johnnyboy2005
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is this true for all cases? i know something can be abelian and not cyclic. thanks
johnnyboy2005 said:is this true for all cases? i know something can be abelian and not cyclic. thanks
A cyclic abelian group is a group in which every element can be generated by repeatedly applying a single element. In other words, every element in a cyclic abelian group can be expressed as a power of a single element.
Some examples of cyclic abelian groups include the group of integers under addition, the group of real numbers under addition, and the group of complex numbers under multiplication.
Yes, all cyclic groups are also abelian. This is because in a cyclic group, the group operation is commutative, meaning the order in which elements are multiplied does not affect the result.
The order of a cyclic abelian group is the number of elements in the group. This is always finite for a cyclic abelian group, as a cyclic group can be generated by a finite set of elements.
Yes, it is true that all cyclic abelian groups have the same structure. This is because any two cyclic abelian groups of the same order are isomorphic, meaning they have the same underlying structure even if the elements are labeled differently.