Exercise 26, § 5

(a) Let $G$ be a group operating on two finite sets $E_1$ and $E_2$. If $s \in G$, let $s_1$ (resp. $s_2$) denote the permutation of $E_1$ (resp. $E_2$) defined by $s$ and $E_1^s$ (resp. $E_2^s$) the set of elements invariant under $s_1$ (resp. $s_2$).

Show the equivalence of the following properties: (i) For all $s \in G$, $\mathrm{Card}(E_1^s) = \mathrm{Card}(E_2^s)$. (ii) For all $s \in G$, the orders of the component cycles of $s_1$ (cf. no. 7, Proposition 7) are the same (to within a permutation) as those of $s_2$. (iii) For all $s \in G$, there exists a bijection $f_s : E_1 \to E_2$ such that $s_2 \circ f_s = f_s \circ s_1$.

If these properties hold, the $G$-sets $E_1$ and $E_2$ are said to be weakly equivalent.

(b) Give an example of two weakly equivalent $G$-sets which are not isomorphic (take $G$ to be a non-cyclic group of order 4).

(c) Let $H_1$ and $H_2$ be two subgroups of a finite group $G$. Show the equivalence of the following properties: (1) For every conjugacy class $C$ of $G$,

(2) The $G$-sets $G/H_1$ and $G/H_2$ are weakly equivalent.

(d) Show that, if $H_1$ and $H_2$ satisfy properties (1) and (2) above and $H_1$ is normal in $G$, then $H_2 = H_1$.

(e) Let $G = \mathfrak{A}_6$ and let

Show that $H_1$ and $H_2$ satisfy properties (1) and (2) and that $H_1$ and $H_2$ are not conjugate in $\mathfrak{A}_6$ (nor in $\mathfrak{S}_6$).