We know that a group can not be written as a union of two proper subgroups and obiously a finite group can be written as a finite union of proper subgroups.So I want to ask if a infinite group be written as a finite union of proper subgroups?Moveover,Can a field be a finite union of proper subfields?
[Math] Can a infinite group be a finite union of proper subgroups
group-theory
Best Answer
Sometimes an infinite group can be written as a finite union of proper subgroups, e.g., every element of ${\bf Z}\oplus{\bf Z}$ is of at least one of the forms $(2a,b)$, $(a,2b)$, or $(a+b,a-b)$ for integers $a,b$, and that gives you 3 proper subgroups whose union is the whole group.
As for fields, note that a field and its subfields are all vector spaces over the prime field, so if a vector space can't be a finite union of proper subspaces, then a field can't be a finite union of proper subfields. The vector space question for infinite fields is handled here. For finite fields, it's easy to see the answer is no. That leaves infinite fields of finite characteristic to think about.