Paper 2, Section II, E

Groups | Part IA, 2020

(a) Let GG be a finite group acting on a finite set XX. For any subset TT of GG, we define the fixed point set as XT={xX:gT,gx=x}X^{T}=\{x \in X: \forall g \in T, g \cdot x=x\}. Write XgX^{g} for X{g}(gG)X^{\{g\}}(g \in G). Let G\XG \backslash X be the set of GG-orbits in XX. In what follows you may assume the orbit-stabiliser theorem.

Prove that

X=XG+xG/Gx|X|=\left|X^{G}\right|+\sum_{x}|G| /\left|G_{x}\right|

where the sum is taken over a set of representatives for the orbits containing more than one element.

By considering the set Z={(g,x)G×X:gx=x}Z=\{(g, x) \in G \times X: g \cdot x=x\}, or otherwise, show also that

G\X=1GgGXg|G \backslash X|=\frac{1}{|G|} \sum_{g \in G}\left|X^{g}\right|

(b) Let VV be the set of vertices of a regular pentagon and let the dihedral group D10D_{10} act on VV. Consider the set XnX_{n} of functions F:VZnF: V \rightarrow \mathbb{Z}_{n} (the integers mod n)\left.n\right). Assume that D10D_{10} and its rotation subgroup C5C_{5} act on XnX_{n} by the rule

(gF)(v)=F(g1v),(g \cdot F)(v)=F\left(g^{-1} \cdot v\right),

where gD10,FXng \in D_{10}, F \in X_{n} and vVv \in V. It is given that Xn=n5\left|X_{n}\right|=n^{5}. We define a necklace to be a C5C_{5}-orbit in XnX_{n} and a bracelet to be a D10D_{10}-orbit in XnX_{n}.

Find the number of necklaces and bracelets for any nn.

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