# Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

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 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.

• 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
• I found this paper link.springer.com/article/10.1007%2FBF01228254 – Charles Hudgins Apr 25 at 18:48

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

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).