Problem Set 4: Products and Maps

Let $G$ be a finite group, let $X, Y$ be $G$-sets, and let $\vp: X \to Y$ be a $G$-map.

1. Prove that $\vp$ maps $G$-orbits in $X$ to $G$-orbits in $Y$.
2. Hence it follows that we can understand all $G$-maps by understanding the $G$-maps between transitive $G$-sets. Let $H, K \subseteq G$ be subgroups and let $g \in G$. Determine the necessary and sufficient conditions for there to exist a $G$-map $\vp : G/H \to G/K$ which sends $H \mapsto gK$.
3. When are $G/H$ and $G/K$ isomorphic as $G$-sets?
4. Recall that an automorphism of an object $X$ is an isomorphism $\vp : X \to X$. The collection $\aut(X)$ of all automorphisms of $X$ forms a group with respect to composition. Prove that the automorphism group of $G/H$ is isomorphic to $N_G(H)/H$ where $N_G(H)$ is the normalizer \[ N_G(H) = \{g \in G : g H g^{-1} = H\}. \]

When we take the product of transitive $G$-sets, the result is typically not transitive anymore. The decomposition of $X \times Y$ tells us the ways in which elements of $X$ and $Y$ can relate to one another, up to symmetry.

1. For example, let $G = S_6$ and let $[6] = \{1,2,\ldots, 6\}$. (In general we write $[n] := \{1, 2, \ldots, n\}$.) Consider $X = \binom{[6]}{2}$ and $Y = \binom{[6]}{3}$ where $X$ is the set of 2 element subsets of $[6]$ and $Y$ is the set of 3 element subsets of $[6]$. Find representatives for each orbit of $S_6$ on $X \times Y$. Given a pair $(x,y) \in X \times Y$, can you succinctly describe the key feature of $x$ and $y$ which determines which orbit it belongs to?
2. Next let $G = D_6$ and let $V$ and $E$ be the sets of vertices and edges of a regular hexagon. These are both $D_6$-sets. Draw a picture to illustrate representatives for the orbits of $D_6$ on $V \times E$.

We know that every transitive $G$-set is isomorphic to $G/H$ for some subgroup $H \subseteq G$. So if $H, K \subseteq G$, how do we decompose the $G$-set $G/H \times G/K$?

1. Prove that every orbit in $G/H \times G/K$ contains an element of the form $(H,gK)$ for some $g \in G$.
2. What is the stabilizer of $(H,gK)$?
3. Given an element $g \in G$ and subgroups $H, K \subseteq G$, the set \[ HgK = \{hgk : h \in H, k\in K\} \] is called a double coset. Let $H\backslash G /K$ denote the set of all double cosets $H g K$. Prove there is a bijection between $H\backslash G /K$ and the orbits of $G$ on $G/H \times G/K$. Be sure to show that your function is well-defined!

1. What are the 0 and 1 elements in $B(G)$? Construct $G$-sets that represent them and explain why they work as additive and multiplicative identities.
2. The isomorphism classes $[G/H]$ corresponding to each conjugacy class of subgroups $H$ form an additive basis for $B(G)$. If $G = S_3$, there are 4 conjugacy classes of subgroups represented by $H = \langle \ve\rangle$, $\langle (12)\rangle$, $\langle (123)\rangle$, and $S_3$ itself. Write \[ 1 = [S_3/S_3], \quad T_2 = [S_3/\langle (123)\rangle], \quad T_3 = [S_3/\langle (12)\rangle], \quad T_6 = [S_3/\langle \ve\rangle], \] so that $T_n$ is the class of a transitive $S_3$-set with $n$ elements, and $1$ is the multiplicative identity from the previous part. Make a multiplication table which shows the product of each pair of elements in our additive basis $1, T_2, T_3, T_6$.

Let $\P$ be the set of all subsets of $[3]$ and let $\G$ be the set of all simple graphs² with vertex set $[3]$ (so $\G$ has $2^3 = 8$ elements, one for each subset of the three possible edges). Both are $S_3$-sets via the relabeling action. Use the notation $T_2, T_3, T_6$ from the previous problem.

1. Express $[\P]$ as a sum of basis elements of $B(S_3)$.
2. Express $[\G]$ as a sum of basis elements of $B(S_3)$.
3. A graph with terminals is a graph on $[3]$ together with a distinguished subset of its vertices (the terminals), that is, an element of $\G \times \P$. Use the multiplication table for $B(S_3)$ to express $[\G \times \P]$ as a sum of basis elements. How many graphs with terminals are there up to relabeling?