Questions tagged [group theory]

CryptoWanderer Wed Sep 18 2024 | 5 answers 1376

Could you please clarify for me if an abelian group is necessarily simple? I understand that an abelian group is a group where the operation is commutative, but I'm unsure if this property alone implies simplicity. Are there any specific conditions or properties that an abelian group must possess in order to be considered simple, or are there examples of abelian groups that are not simple? I'm particularly interested in understanding the relationship between abelian groups and simplicity in the context of group theory.

Daniela Thu Aug 15 2024 | 7 answers 1372

Could you please clarify if the D8 group possesses the property of being abelian? It's important to understand the nature of its operation and how the elements interact under multiplication. Is it true that for any two elements a and b in the D8 group, the product a*b equals b*a? This would indicate that the group is indeed abelian, allowing for a simpler understanding of its structure and behavior. Could you elaborate on this aspect of the D8 group?

Eleonora Wed Aug 14 2024 | 5 answers 1007

I'm curious, can an abelian group ever be classified as simple? I understand that a simple group is one that has no non-trivial normal subgroups, but abelian groups are known for their commutative property. Does this characteristic somehow prevent them from being simple, or are there instances where an abelian group can indeed be considered simple? I'm interested in understanding the mathematical principles behind this question and how they apply to the world of algebra and group theory.

Silvia Wed Aug 14 2024 | 7 answers 1559

Excuse me, could you please elaborate on the concept of "abelian" and then guide me towards understanding its opposite? I'm familiar with abelian groups in mathematics, but I'm curious to know if there's a specific term or concept that describes the properties or structures that are not abelian in nature. Additionally, I'm interested in knowing how these non-abelian structures differ from abelian ones and what implications they may have in various fields, particularly in cryptography and finance where the properties of mathematical groups play a significant role.

Lucia Wed Aug 14 2024 | 5 answers 1404

Excuse me, could you please clarify for me if the group denoted as d4 is indeed not abelian? I understand that in mathematics, an abelian group is one in which the group operation is commutative, meaning that the order of the elements being operated on does not affect the result. So, in the context of d4, which I assume refers to the dihedral group of order 4, is it the case that the multiplication of its elements does not satisfy this commutative property? I'm curious to know if there's a specific reason why d4 is not considered an abelian group.