The lower central series of $S_4$ is given by : $$\gamma_1 =S_4\ge \gamma_2=A_4\ge\gamma_3=A_4\ge \gamma_4 =A_4\ge... $$

This series is clearly central as each $\gamma_i/\gamma_{i+1}$ is central in $S_4/\gamma_{i+1}$ but $S_4$ is known to be not nilpotent. While we know that a lower central series of a group $G$ is central if and only if the group $G$ is nilpotent, which means that the series starts at $G$ and terminates at $\{1\}$ at some finite step. Is there anything I'm missing in this reasoning. Thanks for your help!

  • $\begingroup$ You got the definition slightly wrong: nilpotent iff the lower central series ends at $\{1\}$. $\endgroup$ Sep 23 '18 at 10:13
  • $\begingroup$ that's what I said,nilpotent if the lower central series ends at $\{1\}$ $\endgroup$
    – palio
    Sep 23 '18 at 12:42
  • $\begingroup$ @palio: A finite group is nilpotent iff every maximal subgroup is normal. Since $S_4$ has not-normal maximal subgroups (consider its $2$-Sylow-subgroups), it is not nilpotent. $\endgroup$
    – Tortoise
    Sep 24 '18 at 13:12

A finite group $G$ is nilpotent if and only if the lower central series ends at $1$, if and only if the upper central series ands at $G$. Since $S_4$ has trivial center, it cannot be nilpotent. Consequently, its lower central series cannot and at $1$ - and it doesn't, as you have shown.

  • $\begingroup$ yes I know that, but the lower central series is central which shouldn't be, because it is central if and only if $S_4$ is nilpotent.. $\endgroup$
    – palio
    Sep 23 '18 at 12:43
  • $\begingroup$ My problem is how comes that the lower central series is central while $S_4$ is not nilpotent. $\endgroup$
    – palio
    Sep 23 '18 at 12:45
  • $\begingroup$ In en.wikipedia.org/wiki/Central_series we read that :"The lower central series and upper central series (also called the descending central series and ascending central series, respectively), are characteristic series, which, despite the names, are central series if and only if a group is nilpotent." $\endgroup$
    – palio
    Sep 23 '18 at 12:51
  • 1
    $\begingroup$ Wikipedia has a special definition of "central series", i.e., they define it to terminate. The series for $S_4$ does not terminate, so all is OK. $\endgroup$ Sep 23 '18 at 12:53
  • $\begingroup$ In this document:wwwf.imperial.ac.uk/~jbritnel/Teaching/GTnotes4.pdf the author has the same definition of wikipedia and says on page 30 "The lower central series is only a central series when G is nilpotent. (This, unfortunately, is standard terminology.)" but in page 28 corollary41 he says that $\gamma_i/\gamma_{i+1} $ is central in $G/\gamma_{i+1}$ for all $i$ and then in the next example page 28 he uses the central property to determine $\gamma_3(S_4)$ although $S_4$ is not nilpotent.. I'm confused!! $\endgroup$
    – palio
    Sep 23 '18 at 13:35

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