Questions tagged [abelian group]

RainbowlitDelight Wed Aug 14 2024 | 7 answers 1441

Could you please elaborate on which group of order you are referring to when asking if it is abelian? In mathematics, a group is considered abelian if its operation is commutative, meaning that for any two elements a and b in the group, the result of the operation a applied to b is the same as the result of b applied to a. This property is not inherent to all groups, so it's important to specify the group in question to determine if it is indeed abelian.

Lorenzo Tue Aug 13 2024 | 6 answers 1216

I'm curious about the concept of abelian groups and their potential to be infinite. Can you elaborate on whether or not an abelian group can indeed be infinite in nature? It's intriguing to ponder the implications of an abelian group that doesn't have a finite number of elements, especially in the context of abstract algebra. Could you provide some insight into this idea, perhaps by discussing examples or properties that may suggest the possibility of an infinite abelian group?

Claudio Tue Aug 13 2024 | 7 answers 1406

Could you please clarify for me if an abelian group is indeed closed under its operation? I understand that an abelian group satisfies the properties of associativity, identity, inverses, and commutativity, but I'm specifically wondering about its closure. Does the fact that it is a group inherently mean that it is closed, or is there something specific about an abelian group that guarantees closure? I would appreciate your insight on this matter.

Riccardo Mon Aug 12 2024 | 5 answers 1478

I'm curious to know, does the concept of solvability apply to all abelian groups? Considering that abelian groups possess a certain level of symmetry and simplicity in their structure, does this inherently mean that they can always be decomposed into simpler subgroups in a finite number of steps? Or are there specific conditions or properties that an abelian group must possess in order to be classified as solvable? I'm eager to understand the nuances and implications of this question within the realm of group theory and its applications to cryptography, algebra, and other fields of mathematics.