Does every set have a group structure?

I know that there is no vector space having precisely 6 elements. Does every set have a group structure?

Answer

The trivial answer is “no”: the empty set does not admit a group structure.

The statement

If X is a nonempty set, then there is a binary operation such that (X,) is a group.

is equivalent to the Axiom of Choice.

It is not needed for finite or countable sets: if X is finite, with n elements, then let f:X{0,1,,n1} be a bijection, and use transport of structure to give X the structure of a cyclic group of order n. If X is denumerably infinite, biject with Z and use transport of structure.

For uncountable sets, we can use the Axiom of Choice: let |X|=κ. Then the direct sum of κ copies of Z has cardinality κ, so there is a bijection
f:XiκZ.
Use transport of structure again to make X into a group.

That the converse holds (the statement implies the Axiom of Choice), is proven in this Math Overflow post.

Attribution
Source : Link , Question Author : spohreis , Answer Author : Arturo Magidin

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