I need a little help with the following problem of abstract algebra:

Let $G$ an Abelian group. Clearly, any subgroup of $G$ is normal. Is the opposite true, that is if every subgroup of $G$ is normal, then $G$ is Abelian?


marked as duplicate by Morgan Rodgers, José Carlos Santos, rschwieb, MathematicsStudent1122, Derek Holt abstract-algebra Mar 6 at 17:03

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  • $\begingroup$ Can you add some of your own thoughts to how to proceed? Where are you getting stuck? $\endgroup$ – gt6989b Mar 6 at 16:49
  • $\begingroup$ Have you tried looking at some small nonabelian groups to find a counterexample? Or tried to start a proof? $\endgroup$ – Morgan Rodgers Mar 6 at 16:49
  • $\begingroup$ I see,can you tell me some examples of non-abelian group to check please? $\endgroup$ – José Marín Mar 6 at 16:57
  • $\begingroup$ Please do not title your questions with useless things like "Help with Abstract Algebra". In a case like this, the actual question you are addressing is a good title. $\endgroup$ – rschwieb Mar 6 at 17:01

The converse is not true. Think of the quaternion group of order $8$, $Q=\{\pm 1,\pm i,\pm j,\pm k\}$. Can you see why this gives a counterexample?

  • $\begingroup$ Ok, Let me search about it . Thanks you. $\endgroup$ – José Marín Mar 6 at 17:11
  • $\begingroup$ I don't really recommend searching. The reason is not very difficult. First: what is the possible order of subgroups (Lagrange)? $\endgroup$ – Tortuga Mar 6 at 17:13
  • $\begingroup$ Well, I suppose the posible order of a subgroup is always 2^n where n is the order of the group. $\endgroup$ – José Marín Mar 6 at 17:50
  • $\begingroup$ No. Let $H$ be a subgroup of $G$. Then for sure we must have $|H|\le |G|$. That is the reason we have "sub" here. $H$ must be smaller than $G$ in some sense. Also Lagrange proved that $|H|$ must divides $|G|.$ $\endgroup$ – Tortuga Mar 6 at 17:53
  • $\begingroup$ I see, so, I just have to try for the subgroups wich may have order 1, 2, 4, and 8, rigth? By the way where I can check that things? I am completely new in this stuff. Some good book? to learn and master the material of the abstract algebra. $\endgroup$ – José Marín Mar 6 at 18:06

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