Question

In: Advanced Math

Prove that there is only one possible multiplication table for G if G has exactly 1,...

Prove that there is only one possible multiplication table for G if G has exactly 1, 2, or 3 elements. Analyze the possible multiplication tables for groups with exactly 4 elements, and show that there are two distinct tables, up to reordering the elements of G. Use these tables to prove that all groups with < 4 elements are commutative.

(You are welcome to analyze groups with 5 elements using the same technique, but you will soon know enough about groups to be able to avoid such brute-force approaches.)

Solutions

Expert Solution

If G has exactly one element say then we must have the group identity and so the table is fixed at

*

If G has exactly two elements, one of them must be group identity so we must have

The multiplications must hold and

So we must have the group identity and so the group table is fixed at

*

If G has exactly 3 elements we must have

Then we must have

Consider the element . This can't be equal to since

This means we must have and similarly

And so our table is again fixed as:

*

Colby Messinger answered 1 year ago

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