# $S_4$ is not nilpotent but has central lower central series.

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!

• You got the definition slightly wrong: nilpotent iff the lower central series ends at $\{1\}$. – Nicky Hekster Sep 23 '18 at 10:13
• that's what I said,nilpotent if the lower central series ends at $\{1\}$ – palio Sep 23 '18 at 12:42
• @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. – 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.
• 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.. – palio Sep 23 '18 at 12:43
• My problem is how comes that the lower central series is central while $S_4$ is not nilpotent. – palio Sep 23 '18 at 12:45
• 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. – Dietrich Burde Sep 23 '18 at 12:53
• 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!! – palio Sep 23 '18 at 13:35