Exercise 1, § 1

(a) Show that a group G cannot be the union of two subgroups distinct from G. Show that, for every set I with at least two elements, the commutative group $G = F_2^{(I)}$ is the union of three subgroups distinct from G.

(b) Let $(H_i)_{i \in I}$ be a finite family of subgroups of a group G such that each of the H_i is a subgroup of G of infinite index. Show that G cannot be the union of a finite number of left cosets of the H_i. (Argue by induction on the number of elements in I; if there exist two distinct indices i, j such that the index $(H_i : (H, nH_j))$ is finite, H may be suppressed; if on the other hand $H_i \cap H_j$ is of infinite index in H, for every ordered pair (i, j) of distinct indices, consider an index k such that $H_k$ is maximal in the set of $H_i$ and show that $H_k$ is the union of a finite number of left cosets of the $H_k \cap H_i$ where $i \neq k$.)

(c) Give an example of a commutative ring A and four ideals $a, b_1, b_2, b_3$ of A such that $a \notin b_i$ ($i = 1, 2, 3$), but $a = \bigcup_i b_i$ (use (a)).