How do additive and multiplicative notation differ for groups?

Conditions

• Used for abstract groups.
• The choice of notation changes the symbols for operation and identity but not the underlying group structure.

Reasoning, step by step

1. In additive notation, the operation is written as $a + b$ and the identity as $0$.
2. In multiplicative notation, the operation is written as $a \times b$ (or $ab$) and the identity as $1$.
3. The inverse of $a$ is written as $-a$ in additive notation and $a^{-1}$ in multiplicative notation.
4. The generated subgroup of $y$ in additive notation is $\langle y \rangle = \{\dots, -3y, -2y, -y, 0, y, 2y, 3y, \dots\}$.
5. The generated subgroup of $x$ in multiplicative notation is $\langle x \rangle = \{\dots, x^{-3}, x^{-2}, x^{-1}, 1, x, x^2, x^3, \dots\}$.

Example

The video contrasts: "Additive notation: +, Identity element: 0" and "Multiplicative notation: ×, Identity element: 1".

Common misconceptions

• Assuming that additive notation means the elements are real numbers.
• Thinking that multiplicative notation means the elements are matrices.

Watch the explanation

Connected concepts

Explore next

• Does writing + or × mean the group elements are numbers?
• How is the generated-subgroup definition rewritten in additive notation?

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