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 of the groups in which this property holds?

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

History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{ s.t. } H \cong G/N$, and crossposted this to MO. I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property. After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable.

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    $\begingroup$ Note that all quasisimple groups have the property. $\endgroup$ – Geoff Robinson Dec 30 '12 at 9:58
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    $\begingroup$ There is a discussion of this here: groupprops.subwiki.org/wiki/… $\endgroup$ – Douglas B. Staple Mar 11 '13 at 18:02
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    $\begingroup$ @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 $G/Z(G).$ $\endgroup$ – Geoff Robinson Apr 13 '13 at 17:10
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    $\begingroup$ If $G$ is a free group of rank $\ge 2$ then $G'$ has the required property. $\endgroup$ – Boris Novikov May 18 '13 at 12:07
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    $\begingroup$ I found this paper link.springer.com/article/10.1007%2FBF01228254 $\endgroup$ – Charles Hudgins Apr 25 at 18:48

protected by Community Sep 20 '13 at 13:42

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