June 21st, 2017, 03:09 PM  #1 
Member Joined: Apr 2017 From: India Posts: 73 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: 20,935 Thanks: 2209 
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 
Member Joined: Apr 2017 From: India Posts: 73 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 
Global Moderator Joined: Dec 2006 Posts: 20,935 Thanks: 2209  

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