Can a infinite group be a finite union of proper subgroups?

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?

• How are you going to write $\Bbb Z/3\Bbb Z$ or $\Bbb Z/4\Bbb Z$ as a union of proper subgroups? – Brian M. Scott Aug 15 '12 at 4:37
• Or $\mathbb Z$, for that matter. – Robert Israel Aug 15 '12 at 6:02

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 a side note, it can be shown that a group $G$ can be written as the union of three proper subgroups if and only if it has $H = \mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$ as a quotient, in which case the three subgroups in question are the inverse images (under the quotient map from $G$ to $H$) of the three two-element subgroups of $H$. Gerry's example fits this characterization. – Greg Martin Aug 15 '12 at 8:24
As you say, obviously a finite (noncyclic!) group $G$ can be written as a finite union of proper subgroups - call them $G_1,\dots,G_f$. Now let $H$ be any infinite group. The direct product $G\times H$ can then be written as a finite union of the proper subgroups $G_1\times H,\dots,G_f\times H$.