(4) This exercise outlines a proof that [21 KI 1//IIKİ whenever H and K are subgroups of a group G. (Note that HK-{hk | he H and k E K). The set HK is not always a subgroup of G.) Let -{hK | h є H). Define an action . . H x Ο Ο by the rule hị . ћК hihi. (You may assume that this is an action.) (a) Prove that OH(X). (b) Prove that HK-Hn K. (Here HK denotes the stabilizer of K.) (c) The set O is a partition of HK into lol pieces, where each piece has K elements. Use this fact to write down an equation involving the quantities HKl, Ol, and |K|. (This is a "short answer" question. No justification necessary.) (d) Use the Orbit-Stabilizer Theorem and the above results to establish the equality mentioned at the beginning of the problem. (4) This exercise outlines a proof that [21 KI 1//IIKİ whenever H and K are subgroups of a group G. (Note that HK-{hk | he H and k E K). The set HK is not always a subgroup of G.) Let -{hK | h є H). Define an action . . H x Ο Ο by the rule hị . ћК hihi. (You may assume that this is an action.) (a) Prove that OH(X). (b) Prove that HK-Hn K. (Here HK denotes the stabilizer of K.) (c) The set O is a partition of HK into lol pieces, where each piece has K elements. Use this fact to write down an equation involving the quantities HKl, Ol, and |K|. (This is a "short answer" question. No justification necessary.) (d) Use the Orbit-Stabilizer Theorem and the above results to establish the equality mentioned at the beginning of the problem.