Ian Chen 發問時間： 科學數學 · 6 年前

# 我有數學的問題~急x)

1. under what conditions of a nonempty set S the set of all mappings from S into S forms a group under composition ?

2. let F be a field prove that {a∈F:a≠0} forms a group under the field multiplication.

### 1 個解答

• 6 年前
最佳解答

1.

If S contains at least 2 elements(denoted x and y), then among the set of all mappings from S into S, there exists a mapping which to x associates x and to y associates x as well.

Therefore, this mapping is not injective.

So its inverse doesn't exist, which contradicts the fact that the set to which it belongs should be a group.

Hence, S cannot but contain only one element. And under this condition we can prove easily that the set of all mappings from S into S contains also only one element: the identity fuction, so it's a group.

To conclude, the set of all mappings from S into S forms a group iff S contains only one element.

2.

This property is included in the definition of a field.