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Prove with complete and neat solutions 1. Let H be a proper subgroup of G such that Vx, y ≤ G - H, xy € H. Prove that HAG. 2. Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, then H is normal in G. 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 xy € H ⇒ yx € H, where x, y € G. 4. Consider the additive group Z. Z Prove that Zn for any neZ+. M

Prove with complete and neat solutions 1. Let H be a proper subgroup of G such that Vx, y ≤ G - H, xy € H. Prove that HAG. 2. Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, then H is normal in G. 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 xy € H ⇒ yx € H, where x, y € G. 4. Consider the additive group Z. Z Prove that Zn for any neZ+. M

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 28E: 28. For an arbitrary subgroup of the group , the normalizer of in is the set . a. Prove...
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Prove with complete and neat solutions 1. Let H be a proper subgroup of G such that Vx, y G-H, xy H. Prove that HAG. 2. Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, then H is normal in G. 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 xy € H⇒ yr H, where x, y € G. 4. Consider the additive group Z. Z Prove that Zn for any n € Z+. nZ
Transcribed Image Text:Prove with complete and neat solutions 1. Let H be a proper subgroup of G such that Vx, y G-H, xy H. Prove that HAG. 2. Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, then H is normal in G. 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 xy € H⇒ yr H, where x, y € G. 4. Consider the additive group Z. Z Prove that Zn for any n € Z+. nZ
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