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3. Show that if n ≥ 5, then the only normal subgroups of Sn are {}, An, and Sn. You may use the theorem that An is simple for any n ≥ 5. (Hint: Let HSn. Show that (HnAn) ◄ An, so HnAn = {e} or An. If HnAn An, show that H An or Sn. If HnAn = {ɛ}, use the formula |HA| = |H| · |An\/|H^ An to show that |H| ≤ 2.) = =

3. Show that if n ≥ 5, then the only normal subgroups of Sn are {}, An, and Sn. You may use the theorem that An is simple for any n ≥ 5. (Hint: Let HSn. Show that (HnAn) ◄ An, so HnAn = {e} or An. If HnAn An, show that H An or Sn. If HnAn = {ɛ}, use the formula |HA| = |H| · |An\/|H^ An to show that |H| ≤ 2.) = =

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 17E
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3. Show that if n ≥ 5, then the only normal subgroups of Sn are {}, An, and Sn. You may use the theorem that An is simple for any n ≥ 5. (Hint: Let HSn. Show that (H ^ A₂) ◄ An, so HAn = {ε} or An. If H An = An, show that H = An ^ or Sn. If H An = {}, use the formula |HA| = |H|· |An\/|H^ An to show that |H| ≤ 2.)
Transcribed Image Text:3. Show that if n ≥ 5, then the only normal subgroups of Sn are {}, An, and Sn. You may use the theorem that An is simple for any n ≥ 5. (Hint: Let HSn. Show that (H ^ A₂) ◄ An, so HAn = {ε} or An. If H An = An, show that H = An ^ or Sn. If H An = {}, use the formula |HA| = |H|· |An\/|H^ An to show that |H| ≤ 2.)
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