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 June 21st, 2017, 03:09 PM #1 Newbie   Joined: Apr 2017 From: India Posts: 28 Thanks: 0 Normal Subgroups If H is a subgroup such |G:H|=3, then H is a normal subgroup of G. Well, I have read both Lagrange and Sylow theorems but still I am unable to solve this problem. I know if |G:H|=2, then for sure H is a normal subgroup. Please help me out with this question. It seems tricky to me. Last edited by skipjack; June 21st, 2017 at 03:19 PM.
 June 21st, 2017, 03:19 PM #2 Global Moderator   Joined: Dec 2006 Posts: 18,957 Thanks: 1604 I think H needn't be a normal subgroup of G. Where did you get this problem from? Was any additional information given?
 June 21st, 2017, 11:35 PM #3 Newbie   Joined: Apr 2017 From: India Posts: 28 Thanks: 0 Actually, it was like a true / False statement, So it may happen that it is false,Well in that case what is the reason for falsification of the statement?
 June 22nd, 2017, 02:00 AM #4 Member   Joined: May 2017 From: Russia Posts: 34 Thanks: 5 Symmetric group $\displaystyle S_4$ has a subgroup of order 8, which is not normal.
June 22nd, 2017, 02:40 AM   #5
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Quote:
 Originally Posted by shashank dwivedi I know if |G:H|=2, then for sure H is a normal subgroup.
That is well-known and is proved here. Scroll down to see a counterexample for the case where |G:H|=3.

There is a related theorem given and proved here that you might find interesting.

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