Exercise 12, § 3

Let G be a group, let S be the set of subgroups of G and let F be the set of subsets X of G such that there exists a subgroup H of G and an element $g$ of G such that $X =gH$.

(a) Let $\mu$ be the mapping from $G\times G\times G$ into G defined by $\mu(x, y, z) =xy^{-1}z$. Prove that a nonempty subset X of G belongs to the set F if and only if $\mu(X\times X\times X)\subset X$.

(b) Now prove that, by setting $o(X) =g^{-1}X$ and $t(X) = Xg^{-1}$, for any element $g$ of X, one defines mappings $o: F\rightarrow S$ and $t: F\rightarrow S$.

(c) If $X_1$ and $X_2$ are elements of S such that $t(X_1) =o(X_2)$, set $m(X_1,X_2) = X_1X_2$. Prove that the mapping $m$ is a composition law in the carquois $(S,F, o, t)$ which makes it a groupoid (« Baer groupoid »). It is denoted by Ba(G).

(d) Prove that the orbits of Ba(G) are the conjugacy classes of subgroups of G.

(e) Prove that the isotropy group of Ba(G) at an element H of S is isomorphic to the quotient group $N(H)/H$.