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The Details:
Paraphrasing Robinson's, "A Course in the Theory of Groups (Second Edition)", we have the following
Definition: Let $G$ be a group. Then $G$ is the central product $G_1\circ\dots\circ G_n$ of its normal subgroups $G_i\unlhd G$ if:
- $G=G_1G_2\dots G_n$,
- $[G_i,G_j]=1$ for $i\neq j$, and
- $$G_i\cap \prod_{j\neq i}G_j=Z(G).$$
(Here $Z(G)$ is the centre of $G$.)
This appears to differ, if I'm not mistaken, from contemporary definitions of "central product"; I am using Robinson's because I want to better understand Exercise 5.3.7 from his book (and I have plenty of experience with group presentations).
Definition 2: A group presentation $\langle X\mid R\rangle $ is the quotient of the free group $F_X$ generated by the elements of $X$ by the normal subgroup $\langle\langle R\rangle\rangle$ of $F_X$ generated by $R$, where $R$ is a set of words in $X\cup X^{-1}$; such words can be written as $u=v$ for $uv^{-1}$.
The Question:
Given $H=\langle X_H\mid R_H\rangle$ and $K=\langle X_K\mid R_K\rangle$, find a presentation for the central product $H\circ K$ in terms of $X_H, X_K, R_H, R_K$.
Thoughts:
There's nothing about central products in the standard text for all things to do with group presentations, namely, Magnus et. al's, "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations."
Let $H=\langle X_H\mid R_H\rangle$ and $K=\langle X_K\mid R_K\rangle$. Since, by definition, $[H,K]=1$ in $H\circ K$, I think
$$H\circ K\cong (H\times K)/Q$$
for some $Q\unlhd H\times K$. This is because
$$H\times K\cong \langle X_H\cap X_K\mid R_H\cap R_K\cap\{ hk=kh: h\in X_H, k\in X_K\}\rangle.$$
(A proof of that presentation of $H\times K$ is given here.)
However, I don't know whether the relations $R_H, R_K$ hold for a presentation of $H\circ K$.
Of course, $H\cap K=Z(H\circ K)$.
Please help :)
