# When does an injective group homomorphism have an inverse?

Given two groups $$A$$ and $$B$$ and an injective homomorphism $$f : A \to B$$. When does a homomorphism $$g : B \to A$$ exist with $$g\circ f = \mathrm{id}_A$$ (but not necessarily $$f\circ g = \mathrm{id}_B$$)?

• 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-Peter 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-Peter Stricker Jun 28 '10 at 11:07

## 2 Answers

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 $g$.

If you look at this problem from the side of $$B$$, i. e. how do we find endomorphic retracts of a given group, then there's also somewhat definite answer.

Define equation over group $$G$$ with coefficients in $$H \le G$$ as a collection of elements $$w_i \in H * \mathrm{Free}(x_1, x_2, \dots)$$. Solution of an equation is a homomorphism $$\phi: H * \mathrm{Free}(x_1, x_2, \dots) \to G$$ identical on $$H$$ and such that $$\phi(w_i) = 1$$.

Let's call $$H \le G$$ algebraically closed (should be not confused with algebraic closure of a group, nor by Levine nor Farjoun) if every system of equations $$w_i(X_1, \dots, X_l) = 1$$ with coefficients in $$H$$ have solution in $$G$$ if and anly if it has solution in $$H$$ already.

It's clear that every retract of a group is algebraically closed; сonverse is true if subgroup is f. g. (A. Myasnikov, V. Roman’kov, Verbally closed subgroups of free groups, 2014). Also see this paper about conditions weaker than algebraic closedness https://arxiv.org/pdf/1702.07761.pdf.