Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
Note that all quasisimple groups have the property. –  Geoff Robinson Dec 30 '12 at 9:58
2  
There is a discussion of this here: groupprops.subwiki.org/wiki/… –  Douglas B. Staple Mar 11 '13 at 18:02
1  
@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
1  
If $G$ is a free group of rank $\ge 2$ then $G'$ has the required property. –  Boris Novikov May 18 '13 at 12:07
1  
@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 1

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.

share|improve this answer

protected by Community Sep 20 '13 at 13:42

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.