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June 21st, 2017, 03:09 PM   #1
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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.
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June 21st, 2017, 03:19 PM   #2
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I think H needn't be a normal subgroup of G. Where did you get this problem from? Was any additional information given?
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June 21st, 2017, 11:35 PM   #3
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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?
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June 22nd, 2017, 02:00 AM   #4
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Symmetric group $\displaystyle S_4$ has a subgroup of order 8, which is not normal.
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June 22nd, 2017, 02:40 AM   #5
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Quote:
Originally Posted by shashank dwivedi View Post
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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