Exercise 20, § 7

We retain the notations of the preceding exercise. Let H be a subgroup of G. a) Let R_H be the set of S-connected subsets T of G such that hT ∩ T = ∅ if h ∈ H, h ≠ e. Show that R_H is inductive for the relation of inclusion. b) Let Y be a maximal element of R_H. Show that G = ⋃_{h ∈ H} hY, in other words that Y is a system of representatives of the right classes modulo H. (If G′ = ⋃_{h ∈ H} hY is distinct from G, show that there exist x ∈ G′ and y ∈ Y which are S-neighbors, and deduce that Y ∪ {x} belongs to R_H.) c) Let S_H be the set of elements h ∈ H such that Y and hY are neighbors. It is a symmetric subset of H. Show that an element h ∈ H belongs to S_H if and only if h ≠ e and there exist y, y′ in Y, s ∈ S, such that ys = hy′. Show that S_H generates H. (If H′ is the subgroup of H generated by S_H, one will prove that ⋃_{h ∈ H′} hY is an S-connected component of G.) Recover this result by means of exerc. 14 of I, p. 125. d) Let X_H be a subset of S_H such that S_H is the disjoint union of X_H and X_H⁻¹ (one will show that such a subset exists).

EXERCISES

Show that $X_H$ is a basic family for $H$, and in particular that $H$ is a free group ("Nielsen-Schreier theorem"). (Apply the criterion of exerc. 18 c) of I, p. 146 to $H$, endowed with $S_H$, by observing that two elements $h, h'$ of $H$ are $S_H$-neighbors if and only if $hY$ and $h'Y$ are neighboring subsets of $G$. Use exerc. 19 d) to prove that $S_H$ satisfies condition (ii) of exerc. 18 c).) e) Let $d = (G : H) = \mathrm{Card}(Y)$. Suppose that $d$ is finite. Show that one has:

(Let $T$ be a finite nonempty S-connected subset of $G$, and let $T'$ (resp. $T''$) be the set of pairs of S-neighboring elements whose first element (resp. both elements) belongs (resp. belong) to $T$. We have $\mathrm{Card}(T') = 2\ \mathrm{Card}(X)\mathrm{Card}(T)$ and one will show, by induction on $\mathrm{Card}(T)$, that $\mathrm{Card}(T'') = 2\ \mathrm{Card}(T) - 2$. One will apply the result to $T = Y$, by observing that $S_H$ is equipotent to $T' - T''$.)