2.13 (1966)

Not always. S. I. Adian (Math. USSR–Izv., 5 (1971), 475–484) has constructed a group $A = A(m, n)$ which is torsion-free and has a central element $d$ such that $A(m, n)/\langle d \rangle \cong B(m, n)$, the free $m$-generator Burnside group of odd exponent $n \geqslant 4381$. Given a prime $p$ coprime to $n$, a counterexample with $\pi = \{p\}$ can be found in the form $G = A/\langle d^{p^k} \rangle$ for some positive integer $k$. Indeed, $\langle d \rangle / \langle d^{p^k} \rangle$ is a maximal $p$-subgroup of $G$. Suppose that $A/\langle d^{p^k} \rangle = \langle d \rangle / \langle d^{p^k} \rangle \times H_k/\langle d^{p^k} \rangle$ for every $k$. Then $\langle d \rangle \cap H_k = \langle d^{p^k} \rangle$ for all $k$ and therefore $\langle d \rangle \cap H = 1$, where $H = \bigcap_k H_k$. Since $H$ is torsion-free and is isomorphic to a subgroup of $A/\langle d \rangle \cong B(m, n)$, we obtain $H = 1$. This implies that $A$ is isomorphic to a subgroup of the Cartesian product of the abelian groups $A/H_k$, a contradiction. (Yu. I. Merzlyakov, 1973.)