RO-NMO 2005 Problem 122

Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$ , for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$ ).

a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$ ; b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$ .

Calin Popescu