Does writing + or × mean the group elements are numbers?

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. Recognize that $+$ and $\times$ are symbols for the group operation.
2. Understand that $0$ and $1$ are symbols for the identity element.
3. Realize that these symbols do not imply the elements are real numbers.
4. Examples of non-numeric groups include symmetries of a polygon or permutations.

Example

The lecturer warns: "keep thinking abstractly even though the notation may lure the mind toward real numbers."

Common misconceptions

• Assuming that because a group is written additively, it must be a subgroup of $(\mathbb{R}, +)$.
• Thinking that multiplicative notation implies the elements are invertible matrices.

Watch the explanation

Connected concepts

Explore next

• How do additive and multiplicative notation differ for groups?
• What is $\langle x \rangle$ and how is it built from $x$?

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