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Let H and K be subgroups of a group G. (a) Define HK = {hk | he H, ke K}. Show that if K is normal in G, then HK ≤G. (b) Show that if H and K are normal in G, then HK is normal in G. (c) Show that H is normal in G if and only if xy € H ⇒ yx € H, where x, y € G.

Let H and K be subgroups of a group G. (a) Define HK = {hk | he H, ke K}. Show that if K is normal in G, then HK ≤G. (b) Show that if H and K are normal in G, then HK is normal in G. (c) Show that H is normal in G if and only if xy € H ⇒ yx € H, where x, y € G.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 19E
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Prove with complete and neat solutions 3. Let H and K be subgroups of a group G. (a) Define HK = {hk | h H, k € K}. Show that if K is normal in G, then HK ≤G. (b) Show that if H and K are normal in G, then HK is normal in G. (c) Show that H is normal in G if and only if ry H⇒ yx € H, where x, y € G.
Transcribed Image Text:Prove with complete and neat solutions 3. Let H and K be subgroups of a group G. (a) Define HK = {hk | h H, k € K}. Show that if K is normal in G, then HK ≤G. (b) Show that if H and K are normal in G, then HK is normal in G. (c) Show that H is normal in G if and only if ry H⇒ yx € H, where x, y € G.
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