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LagrangeEuler
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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.LagrangeEuler said:Is it mathematically correct to call any group operation in ##(G,\cdot)## composition?
Yes, although composition is used in some other contexts, e.g. composition series. So product orLagrangeEuler 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
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
As this it's at least the operation in ##GL(n)##.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.
Group theory is a branch of mathematics that studies the algebraic structures called groups. A group is a set of elements that can be combined together using a binary operation (such as addition or multiplication) and satisfies certain properties, including closure, associativity, identity, and inverse.
The composition of a group refers to the operation used to combine its elements. This operation can be addition, multiplication, or any other binary operation that satisfies the group properties. It is denoted by the symbol * or · and is read as "composed with" or "multiplied by".
Group composition differs from regular arithmetic in that the elements being combined do not necessarily have to be numbers. They can be any objects that satisfy the group properties, such as matrices, functions, or even abstract concepts. Additionally, the composition operation in a group may not follow the same rules as addition or multiplication in regular arithmetic.
The identity element in group theory is the element that, when combined with any other element in the group using the composition operation, results in the same element. For example, in the group of integers under addition, the identity element is 0, since adding 0 to any integer does not change its value.
Group theory has many applications in physics, chemistry, and other branches of science. For example, it is used to study the symmetries of molecules in chemistry, the properties of subatomic particles in particle physics, and the behavior of waves in optics. It also has applications in computer science, cryptography, and music theory.