Paper 3, Section II, D

(a) State and prove Lagrange's theorem.

(b) Let $G$ be a group and let $H, K$ be fixed subgroups of $G$. For each $g \in G$, any set of the form $H g K=\{h g k: h \in H, k \in K\}$ is called an $(H, K)$ double coset, or simply a double coset if $H$ and $K$ are understood. Prove that every element of $G$ lies in some $(H, K)$ double coset, and that any two $(H, K)$ double cosets either coincide or are disjoint.

Let $G$ be a finite group. Which of the following three statements are true, and which are false? Justify your answers.

(i) The size of a double coset divides the order of $G$.

(ii) Different double cosets for the same pair of subgroups have the same size.

(iii) The number of double cosets divides the order of $G$.

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