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I'm not very good with English terms of group theory but here is the question :

$\forall H\trianglelefteq G \rightarrow \exists H' \trianglelefteq G : {G\over H} \approx H'$

is above statement always true? or should there be some other constraints?

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    @wj32 I've just noticed that, but what about the constraints to make that statement true? is M.K.'s suggestion the weakest constraints we can find?2012-11-03

2 Answers 2

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This is not true in general. The smallest counterexample can be found in the quaternion group $Q_8$. There $Q_8/Z(Q_8)$ is isomorphic to the Klein $4$-group $\mathbb{Z}_2 \times \mathbb{Z}_2$, but every subgroup of order $4$ in $Q_8$ is cyclic.

However, if we assume that $G$ is finite and abelian, then the statement is true.

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    I think this is still a strong example, because it shows that $G/H$ not only does not have to be isomorphic to a normal subgroup of $G$, but $G/H$ does not have to be isomorphic to a subgroup of $G$ at all (a common mistake).2012-11-03
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Hint: The symmetric group $S_5$ has exactly three normal subgroups: $\{1\}$, $S_5$ and the alternating group $A_5$, which has index 2 in $S_5$.

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    What on earth is there to downvote about this?2012-11-05