Paper 3, Section II, D

Groups | Part IA, 2015

(a) State and prove Lagrange's theorem.

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

Let GG 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 GG.

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

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

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