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If $G$ is a group and $S\subseteq G$, let $\langle S \rangle$ be the intersection of all subgroups of $G$ containing $S$.

Let $S_\omega$ denote the group of all bijections $f:\omega\to\omega$ with composition.

Is there $M\subseteq S_\omega$ such that $\langle M \rangle =S_\omega$, but for all $m\in M$ we have $\langle M\setminus\{m\} \rangle\neq S_\omega$?

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1 Answer 1

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No.

Indeed, F. Galvin proved in 1995 that every countable subset of $S_\omega$ is contained in a finitely generated subgroup (and also $S_\kappa$ for every infinite $\kappa$). By contradiction suppose $M$ exists. Let $I$ be an infinite countable subset of $M$, so $I\subset \langle F\rangle$ for some finite $F$, and hence there exists a finite subset $J$ of $M$ such that $F\subset \langle J\rangle$. Hence, for $g\in I\smallsetminus J$, we have $g\in\langle J\rangle$, and therefore $M\smallsetminus\{g\}$ generates $S_\omega$.

More generally no such $M$ exists in any uncountable group $G$ with uncountable cofinality. Indeed fix $I\subset M$, write finite subsets $I_0\subset I_1\subset I_2...$ with union $I$, and define $M_n=(M\smallsetminus I)\cup I_n$. Then $G =\bigcup\langle M_n\rangle$ (increasing union). By definition of uncountable cofinality, $G=\langle M_n\rangle$ for some $n$: contradiction. Thus this shows that for every generating subset $M$, there exists $M'\subset M$ with $M\smallsetminus M'$ infinite, such that $M'$ generates $G$.

This also shows some stronger consequence: $G$ is not "infinitely independently generated": there is no sequence $(S_n)_{n\in\omega}$ of subsets of $G$ such that $\bigcup_k S_k$ generates $G$, but $\big\langle\bigcup_{k\neq n}S_k\big\rangle\neq G$ for every $n$. The latter condition has the advantage of being purely intrinsic to the poset of subgroups of $G$.

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  • $\begingroup$ Brilliant - thanks @YCor! $\endgroup$ Mar 21, 2020 at 9:20
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    $\begingroup$ Just about the digression at the end: the group $\mathbf{Q}$ has no minimal generating subset, but yet is infinitely independently generated (because it admits a quotient that is an infinite direct sum). $\endgroup$
    – YCor
    Mar 21, 2020 at 12:32

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