Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / B2

The orbit of $s$ is $O_s=\{gs:g\in G\}$, and its stabilizer is $G_s=\{g\in G:gs=s\}$. The identity fixes $s$; if $g,h$ fix it then $(gh)s=g(hs)=s$; and $gs=s$ implies $g^{-1}s=s$. Thus $G_s$ is a subgroup.

A group action is transitive when it has just one orbit: for every $s,t\in S$ there is $g\in G$ with $gs=t$.

On the set of left cosets $G/H$, define $g\cdot (xH)=(gx)H$. Given $xH$ and $yH$, the element $y{x}^{-1}$ sends the first to the second. Thus the coset action is transitive. In particular the orbit of $H$ is the entire coset space, and its stabilizer is $H$.

Two group actions on $S_1,S_2$ are equivalent if there is a bijection $\varphi :S_1\to S_2$ such that $\varphi (gs)=g\varphi (s)$ for all $g,s$. Such a bijection is equivariant.

The orbit-stabilizer theorem in this form says that the action on $O_s$ is equivalent to the left coset action on $G/G_s$, by $g{G}_s\mapsto gs$. This map is well-defined and injective because $gs=hs$ exactly when $h^{-1}g\in G_s$, and it is onto by the definition of the orbit. It is equivariant by multiplication.

Thus every transitive action is a coset action. Two such actions $G/H$ and $G/K$ are equivalent exactly when $H,K$ are conjugate: stabilizers at different points of one orbit are conjugate, and a conjugacy explicitly supplies a coset-action equivalence. This last classification is what makes counting by orbit types possible.

A cyclic group of order six has exactly one subgroup of each order $1,2,3,6$, and no others. Since it is abelian, these are also its four conjugacy classes of subgroups. Thus there is one transitive group action of each size $1,2,3,6$.

An action on six points is a disjoint union of its orbits. Its equivalence type is therefore determined by nonnegative integers satisfyingIf $n_6=1$ there is one possibility. If $n_6=0$, the possibilities $n_3=0,1,2$ allow respectively $4,2,1$ choices of $n_2$, with $n_1$ then fixed. HenceThis is counting finite group actions by transitive orbit types. Equivalently, a generator acts as a permutation whose cycles have lengths dividing six, and these are its eight possible cycle types on six points.

The symmetric group $S_3$ has subgroups of orders $1,2,3,6$. There is one trivial subgroup, three conjugate subgroups generated by transpositions, one order-three subgroup $A_3$, and the whole group. These are all subgroups by Lagrange's theorem. Thus again there is one conjugacy class for each index $6,3,2,1$.

The coset action classification in part (c) supplies exactly one transitive orbit type of each of these sizes. The same orbit-multiplicity equation as in part (i) applies, givingThe equality of the two counts comes from their identical lists of transitive orbit types, rather than from the groups being isomorphic.