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Normal Subgroups are subgroups where all left cosets are right cosets. For abelian groups all subgroups are normal.

I want to discuss about a non-abelian group whose subgroups are all normal. Please give an example.

Can we give example of a finite non-abelian group with same property ?

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  • $\begingroup$ Lots of the very first examples of groups we know have normal subgroups. What non-abelian groups do you know? $\endgroup$ – Tyler Oct 22 '13 at 13:57
  • $\begingroup$ See also mathoverflow.net/questions/25307/… $\endgroup$ – lhf Jan 16 at 9:58
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The quaternion group is a finite, nonabelian group where every subgroup is normal.

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  • $\begingroup$ And as the linked page mentions, the term for groups whose subgroups are all normal are called Dedekind groups. They also use "Hamiltonian" for not-abelian Dedekind groups, but I feel like naming schemes like that seem to run against the grain of 20th century mathematics ideas... $\endgroup$ – rschwieb Oct 22 '13 at 13:58
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If $H$ is a finite subgroup of a group $G$ then it has only finitely many conjugates $k^{-1}Hk$ and these all have finite index in $G$ also. Taking the intersection of these subgroups, $N=\cap_{k\in G}k^{-1}Hk$ yields a normal subgroup which also has finite index.

This is a constructive method of creating normal subgroups. However, if $G$ is finite then the subgroup $N$ may be trivial. Note that if $G$ is infinite and simple then the above working implies that $G$ contains no subgroups of finite index.

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