On periodic groups saturated by finite alternating groups
We prove that every countable residually finite periodic group embeds in a countable simple periodic group in which every finite subgroup is contained in a finite alternating subgroup. The construction uses finite regular wreath products and a diagonal sequence of embeddings. Applied to the Gupta--Sidki $3$-group, it produces a simple periodic group which is not locally finite, thereby giving a negative answer to Problem~20.57 of the \emph{Kourovka Notebook}. In the resulting group, two finite alternating subgroups can generate an infinite subgroup.
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Lemma 2.3
Let $Q$ be a finite group, and put $m=2|Q|+5$. There is an embedding $\iota:Q\longrightarrow\Alt(m)$. If $N$ is any group, then $\iota$ induces an embedding \[ \widehat\iota:N\wrp Q\longrightarrow N\wrp\Alt(m) \] whose restriction to the natural copy of $Q$ is $\iota$.
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- External referenceD. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996.LaTeX source\bibitem{Robinson} D.~J.~S. Robinson, \emph{A Course in the Theory of Groups}, 2nd ed., Graduate Texts in Mathematics, vol.~80, Springer-Verlag, New York, 1996.
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Machine-readable theorem index · 11 statements
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- PF:2026.000024/v1/THM-1.1
Let $H$ be a countable residually finite periodic group. Then $H$ is isomorphic to a subgroup of a countable nonabelian simple periodic group $G$ such that every finite subgroup of $G$ is contained in a subgroup isomorphic to $\Alt(m)$ for some $m\geq5$.
- PF:2026.000024/v1/COR-1.2
By Theorem 1.1, there exists a countable non-locally finite periodic group saturated by finite alternating groups. Such a group can moreover be chosen to be simple.
- PF:2026.000024/v1/LEM-2.1
Let $P$ be a group, let $N\trianglelefteq P$ have finite index, and put $Q=P/N$. Write $\pi:P\longrightarrow Q$ for the canonical homomorphism. If $F\leq P$ satisfies $F\cap N=1$, then there exists a section $r:Q\longrightarrow P$ of $\pi$ such that \[ r(1)=1 \quad\text{and}\quad r(x\pi(f))=r(x)f \qquad(x\in Q,\ f\in F). \]
- PF:2026.000024/v1/LEM-2.2
Let $P$, $N$, $Q$ and $F$ be as in Lemma 2.1. There exists an embedding \[ \theta:P\longrightarrow N\wrp Q \] such that \[ \theta(F)\leq Q \quad\text{and}\quad \operatorname{pr}_Q\circ\theta=\pi, \] where $\operatorname{pr}_Q:N\wrp Q\longrightarrow Q$ denotes the natural projection.
- PF:2026.000024/v1/LEM-2.3
Let $Q$ be a finite group, and put $m=2|Q|+5$. There is an embedding $\iota:Q\longrightarrow\Alt(m)$. If $N$ is any group, then $\iota$ induces an embedding \[ \widehat\iota:N\wrp Q\longrightarrow N\wrp\Alt(m) \] whose restriction to the natural copy of $Q$ is $\iota$.
- PF:2026.000024/v1/LEM-2.4
Let $N$ be a periodic residually finite group and let $A$ be finite. Then $N\wrp A$ is periodic and residually finite. If in addition $N$ is countable, so is $N\wrp A$.
- PF:2026.000024/v1/PROP-2.5
Let $P$ be a countable residually finite periodic group, and let $F\leq P$ be finite. There exist a countable residually finite periodic group $P^+$, an embedding $\varphi:P\longrightarrow P^+$, and a subgroup $K\leq P^+$ such that \[ \varphi(F)\leq K\simeq\Alt(m) \qquad\text{for some }m\geq2|F|+5. \]
- PF:2026.000024/v1/PROP-3.1
Let $H$ be a countable residually finite periodic group. Then there exists a countable periodic group $G$ containing an isomorphic copy of $H$ such that, for each finite subgroup $F\leq G$, there is a subgroup $K\leq G$ with F\leq K\simeq\Alt(m), \qquad m\geq2|F|+5.
- PF:2026.000024/v1/LEM-3.2
Let $G\ne1$ be a periodic group such that every finite subgroup of $G$ is contained in a subgroup isomorphic to $\Alt(m)$ for some $m\geq5$. Then $G$ is nonabelian and simple.
- PF:2026.000024/v1/COR-4.1
The group $G$ in Corollary 1.2 can be chosen to contain two finite nonabelian simple alternating subgroups $K$ and $L$ such that $\langle K,L\rangle$ is infinite.
- PF:2026.000024/v1/COR-4.2
The group $G$ in Corollary 1.2 may be chosen so that it contains an isomorphic copy of every finite group.