Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G\: \text{ s.t. } \:H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this property fails?

If this question is too broad, I might ask if such a characterization exists for $p$-groups.

As @Ralph points out, we could restate this property as "groups in which at least one subgroup is not an endomorphism kernel." Evidently there is a bit of literature on groups for which the only endomorphism kernels are trivial or the whole group, but (outside of simple groups) this is a substantially stronger condition.

Update: I crossposted to MO and received an answer to the (now omitted) peripheral question about probability. The characterization is still unanswered; however, most finite groups appear to have this property, so it may be a better idea to instead look for a characterization of finite groups satisfying the opposite condition, that is, $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$. (Of course, this would also provide a good enough answer to the original question.)

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Note that all quasisimple groups have the property. –  Geoff Robinson Dec 30 '12 at 9:58
There is a discussion of this here: groupprops.subwiki.org/wiki/… –  Douglas B. Staple Mar 11 '13 at 18:02
@Alexander: However, a genuinely quasisimple group $G$ , that is one with a non-trivial center, has a homomorphic image which is not a subgroup, namely $G/Z(G).$ –  Geoff Robinson Apr 13 '13 at 17:10
If $G$ is a free group of rank $\ge 2$ then $G'$ has the required property. –  Boris Novikov May 18 '13 at 12:07
@yatima2975 I don't think that is in fact a restatement unless every subgroup $H$ can be put into a subnormal series. This won't be the case in general, for instance in any finite non-nilpotent group. –  Avi Steiner May 22 '13 at 6:11

1 Answer

If you were to change your question to that of characterizing all groups $G$ with $N⊴G$ such that there is no subgroup $H$ of $G$ with $H≅G/N$ and $H \cap N =1$, then you'd be asking for a characterization of all nonsplit extensions of $N$ by $H$. This is part of the "extension problem" which is intractable. The way the question is presently stated, the desired characterization would be a subclass of the class of all nonsplit extensions.

For example, your characterization would include all groups $G=NH$ where $H$ is a simple group with nontrivial Schur multiplier $N$. Here $G$ is a nonsplit (central) extension of $N$ by $H$, but $G$ contains no subgroup isomorphic to $H$. However, as presently stated, your characterization would not include the quaternion group $Q_8$, even though $Q_8$ is a nonsplit extension of $\langle i\rangle\cong \mathbb Z_4$ by $\mathbb Z_2$. Indeed, $Q_8$ contains the subgroup $\langle -1\rangle\cong\mathbb Z_2$. Note, however, that $\langle -1\rangle$ is contained in the kernel of the quotient map $Q_8\to Q_8/\langle i\rangle$, so it would not violate the "amended" version of the question I mentioned above.

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protected by Community♦Sep 20 '13 at 13:42

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