MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given two groups A and B and an injective homomorphism f: A → B. When does a homomorphism g: B → A exist with gf = idA (but not necessarily fg = idB)?

share|cite|improve this question
You may assume that $f$ is the inclusion of a subgroup $A \subseteq B$. I don't think that there is a general simple criterion. If $B$ is abelian, you have the splitting lemma which says that $A \subseteq B$ has a retract if and only if $B \to B/A$ has a section if and only if $A$ is a direct summand of $B$. – Martin Brandenburg Jun 28 '10 at 10:49
BB in his answer below mentions semidirect products, you talk about direct sums. Who of you is right (or both)? – Hans Stricker Jun 28 '10 at 10:57
I only adressed the case that $B$ is abelian. – Martin Brandenburg Jun 28 '10 at 11:05
@Martin: I see. – Hans Stricker Jun 28 '10 at 11:07
up vote 7 down vote accepted

If and only if $B$ is a semidirect product of $A$ and another group (the latter is normal). One direction is obvious, and another direction is easy: $B$ is a semidirect product of the image of $f$ and the kernel of $g$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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