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I'm trying to solve the problem 2.10.3 from Artin's Algebra first edition:

Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P which contains 1. Prove that N is a normal subgroup of G and that P is the set of its cosets.

I manage to prove that N is a group, that is normal, and that is't cosets are contained in the elements of the partition but I don't see how to prove the converse inclusion, any hint?

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    $\begingroup$ So you have $Ng \le P_1$ for some partition element $P_1$, $Ng^{-1} \le P_2$, giving $P_1P_2 \le N$, hence $P_1Ng^{-1} \le N$, so $P_1 \le Ng$. $\endgroup$ – Derek Holt Sep 30 '15 at 16:06
  • $\begingroup$ That's it. Thanks Derek! $\endgroup$ – Jose Fernandez Sep 30 '15 at 17:25
  • $\begingroup$ .... so $P_1 N \le Ng$? @DerekHolt $\endgroup$ – athos Apr 22 '16 at 9:12
  • $\begingroup$ @athos I don't understand what you are asking. I gave a hint which enabled Jose Fernandez to complete huis solution. $\endgroup$ – Derek Holt Apr 22 '16 at 16:15
  • $\begingroup$ @DerekHolt I guess you comment has an typo at the end, "so $P_1 \le Ng$" might be "so $P_1 N \le Ng$"? $\endgroup$ – athos Apr 23 '16 at 1:21

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