Is Group Operation in (G,*) Considered Composition in Mathematics?

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Discussion Overview

The discussion revolves around whether it is mathematically correct to refer to any group operation in the context of a group (G, *) as composition. The scope includes theoretical considerations of mathematical terminology and notation related to group operations.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants suggest that as long as consistency is maintained, it is acceptable to call any group operation composition, regardless of the specific notation used.
  • Others argue that while it may be technically correct, the term "composition" is more appropriate in specific contexts, such as with functions, and that terms like "product" or "sum" might be clearer in other cases.
  • A participant expresses a strong disagreement, stating they would never refer to the operation of addition (e.g., 3 + 5 = 8) as composition, emphasizing that composition should primarily be associated with functions.
  • Another participant supports this view, indicating that composition is typically used in contexts like function composition rather than for operations in groups like GL(n).

Areas of Agreement / Disagreement

Participants do not reach a consensus on the terminology, with multiple competing views on whether the term "composition" is appropriate for group operations. Some participants advocate for its use, while others reject it in favor of more specific terms.

Contextual Notes

There is a lack of clarity regarding the definitions and contexts in which the term "composition" should be applied, particularly in distinguishing between operations in groups and function composition.

LagrangeEuler
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Is it mathematically correct to call any group operation in ##(G,\cdot)## composition?
 
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LagrangeEuler said:
Is it mathematically correct to call any group operation in ##(G,\cdot)## composition?
As long as you stay consistent, yes. Usually one wouldn't describe groups like ##\mathbb{Z}## by a dot as binary operator, or a matrix group like ##GL(n)## by an addition, but strictly speaking it doesn't matter which symbol for the operation is used. This changes however, if more than one operation is involved. So you may write ##a \circ b\, , \,a+b\, , \,a \cdot b\, , \,a * b## or simply ##ab##. But you should think about the readability: ##3 \cdot 5 = 8## in ##(\mathbb{Z},+)## might be quite disturbing.
 
Thanks. I do not asked about notation but just calling. Is it fine to say that 3+5=8 composition of numbers 3 and 5 is 8
 
LagrangeEuler said:
Thanks. I do not asked about notation but just calling. Is it fine to say that 3+5=8 composition of numbers 3 and 5 is 8
Yes, although composition is used in some other contexts, e.g. composition series. So product or addition sum might be better, but on principle: yes.
 
LagrangeEuler said:
Thanks. I do not asked about notation but just calling. Is it fine to say that 3+5=8 composition of numbers 3 and 5 is 8

No, I would never call that composition. I have never seen it referred to as composition. Composition should be used for functions mainly.
 
micromass said:
No, I would never call that composition. I have never seen it referred to as composition. Composition should be used for functions mainly.
As this it's at least the operation in ##GL(n)##.
 

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