# injectivity of pushout?

We have the following pushout diagram: $$\begin{array}{ccc} \langle X, Y \rangle & \xrightarrow{\alpha} & \mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d \\ \downarrow \scriptsize{\beta} && \downarrow \scriptsize{g} \\ \mathbb{Z}_e & \xrightarrow{f} & G\end{array}$$ Suppose that $$\alpha$$ is injective, and $$\beta$$ maps $$X,Y$$ to $$1\in \mathbb{Z}_e$$. ($$\mathbb{Z}_n$$ means the cyclic group of order $$n$$.)

I wonder that in which cases, $$f$$ is injective. For any positive integers $$a,b,c,d,e \geq 2$$? Otherwise, only for relative prime or distinct prime $$a,b,c,d,e$$?

(I found that if $$\phi$$ is injective, then $$\psi$$ is injective in "adhesive" categories. The category of abelian groups is adhesive, and the category of groups isn't. I also wonder which subcategory of groups are adhesive. ) $$\begin{array}{ccc}A & \xrightarrow{\phi} & B \\ \downarrow && \downarrow \\ C & \xrightarrow{\psi} & D\end{array}$$.

In the category of groups, there is a counterexample: http://math.stackexchange.com/questions/601463/a-monomorphism-of-groups-which-is-not-universal

I can change my diagram similarly to the above counterexample: $$\begin{array}{ccc}\mathbb{Z} & \xrightarrow{\alpha'} & (\mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d)/\langle X=Y \rangle \\ \downarrow \scriptsize{\beta'} && \downarrow \scriptsize{g'} \\ \mathbb{Z}_e & \xrightarrow{f} & G\end{array}$$ $$\alpha'(1)=X$$, any words in $$X,Y$$ are nontrivial in $$\mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d$$, and $$\beta'$$ is surjective. Then, $$\alpha'$$ is also injective. For instance, fix $$Y=1_a 1_b 1_c 1_d$$ with a generator $$1_n$$ of $$\mathbb{Z}_n$$. May $$f$$ not be a monomorphism?

• The pushout of a one-to-one map in the category of sets is one-to-one. In the category of groups, my guess is that you have to add the free compositions. So I would be surprised that $f$ is not one-to-one. Commented May 6, 2020 at 9:54
• What has this got to do with topos theory ? Commented May 6, 2020 at 11:55
• @MaximeRamzi I found that the above injectivity holds in any topos (toposes are adhesive). I cannot understand what topos is, but I think it may provide some answer. If topos theory has nothing to do with my question, sorry. Then, tell me. I will subtract it.
– qkqh
Commented May 6, 2020 at 16:44
• @PhilippeGaucher What does it mean to add the free compositions in the category of groups? Could you explain it in detail?
– qkqh
Commented May 6, 2020 at 16:52
• @qkqh I've learnt something today, I could not imagine that there would be a counterexample. In my mind, a group is a one-object category, and the pushout must contain all possible compositions. Commented May 6, 2020 at 21:29

The first exercise in Serre’s book Trees spells this out (I’ve corrected a missing “normal”):

Let $$f_1\colon A \to G_1$$ and $$f_2\colon A \to G_2$$ be two homomorphisms and let $$G$$ be their pushout. We define subgroups $$A^n$$, $$G^n_1$$ and $$G^n_2$$ of $$A$$, $$G_1$$ and $$G_2$$ recursively by the following conditions:

• $$A^1 = \{1\}, \qquad G^1_1=\{1\}, \qquad G^1_2 = \{1\}$$

• $$A^n =$$ normal subgroup of $$A$$ generated by $$f^{-1}_1(G^{n-1}_1)$$ and $$f^{-1}_2(G^{n-1}_2)$$

• $$G^n_i =$$ normal subgroup of $$G_i$$ generated by $$f_i(A^n)$$.

Let $$A^\infty$$, $$G^\infty_i$$ be the unions of the $$A^n$$, $$G^n_i$$, respectively. Show that $$f_i$$ defines an injection $$A/A^\infty \to G_i/G^\infty_i$$ and that $$G$$ may be identified with the amalgam of $$G_1/G^\infty_1$$ and $$G_2/G^\infty_2$$ along $$A/A^\infty$$.

It follows (using the results of no. 1.2 in Trees) that the kernel of $$A\to G$$ is $$A^\infty$$ and that the kernel of $$G_i \to G$$ is $$G_i^\infty$$.

• Thank you very much!! Applying the theory to my lower diagram ($f_1:=\alpha', f_2=\beta'$), $A^n=A^\infty=e\mathbb{Z}, G_1^n=G_1^\infty=\langle \langle X^e \rangle \rangle, G_2^n=G_2^\infty=1$ for any $n>1$, and so $A/A^\infty=\mathbb{Z}_e \to G_1/G_1^\infty=G$ is injective. Hence, $f$ is alway injective!! Wow! Could you tell me some references to study it?
– qkqh
Commented May 11, 2020 at 8:51
• I want to insert missing subindex "1" on the codomain of $f_1$ in your answer, but I couldn't. I should fix at least 6 characters.
– qkqh
Commented May 11, 2020 at 8:55
• I've found that it is a book. However, I cannot find the nomality condition that you've modified.
– qkqh
Commented May 11, 2020 at 9:24
• In the exercise as written Serre does not say that $A^n$ and $G^n_i$ should be the normal subgroups generated by the conditions above. I spent a full day confused about this; I seem to recall that one family of subgroups was likely normal, but couldn't see any reason for the other family to be normal. Commented May 13, 2020 at 4:23
• I remember getting stumped by this omission of normal as well. Serre does say dist. (for distingué) in the French original so it was lost in (the generally excellent) translation. Commented May 14, 2020 at 12:42