8. Let (G,*) be a group, and let H, K be subgroups of G. DefineH* K = {hk: he H, ke K}.Show that H K≤G⇒H✩K= K ★ H.9. Show that (Z12, +) is a cyclic group. Find the number of its genera-

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Answered: 8. Let (G,*) be a group, and let H, K…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 25E: If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.

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8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H* K≤G⇒H✩K= K★ H. 9. Show that (Z12, +) is a cyclic group. Find the number of its genera-

8. Let (G,) be a group, and let H, K be subgroups of G. Define HK={hk: he H, ke K}. Show that H K≤G⇒H✩K= K★ H. 9. Show that (Z12, +) is a cyclic group. Find the number of its genera-