Are $p$-groups Engel groups?

Question

A group $G$ is termed Engel if whenever $x, y \in G$, there exists an integer $n$ (depending on $x$ and $y$ such that $[x,y,\dots,y]=1$, where $y$ occurs $n$ times.

Is it true that every infinite $p$-group is Engel?

This question is related to http://math.stackexchange.com/questions/431660/finite-p-groups-are-nilpotent — Yassine Guerboussa, Jun 30 '13 at 15:54

score 5 · Answer 1 · answered Jul 01 '13 at 14:43

No. There are infinite $p$-groups that are not Engel groups.

Baer (1957) showed that in groups in which every ascending chain of subgroups eventually stabilizes (“Noetherian groups”), the set of left Engel elements is the Fitting subgroup and the set of right Engel elements is the hypercenter. In particular, this generalizes Zorn's theorem (that a finite Engel group is nilpotent) to Noetherian groups (Noetherian+Engel implies nilpotent).

Since a Tarski monster is fairly trivially a Noetherian group (maximal chains of subgroups are $1 < \langle x \rangle < G$) the only left or right Engel element is the identity. Since a Tarski monster has exponent $p$, it is a $p$-group.

Be cautious that there are many open problems about Engel elements in general. Early results such as Baer's are very nice, but the later research is half theorem and half counterexample, as the behavior of Engel sets is difficult to predict.

This paper also has an important result in finite groups, the Baer-Suzuki theorem.

Baer, Reinhold. “Engelsche elemente Noetherscher Gruppen.” Math. Ann. 133 (1957), 256–270. MR86815 DOI:10.1007/BF02547953

I'm really grateful, many thanks Jack Scmidt and DonAntonio. — Yassine Guerboussa, Jul 01 '13 at 15:29

score 2 · Accepted Answer · answered Jun 30 '13 at 16:26

2

Well, nilpotent groups are almost trivially $\;n-$ Engel, with $\;n=$ the group's nilpotency class. Since finite $\;p-$ groups are nilpotent then they are Engel groups.

About infinite $\;p-$groups I can't be sure, but I think in general the answer is negative since there are simple (infinite) $\;p-$groups...

answered Jun 30 '13 at 16:26

DonAntonio

214,715

Thanks for the answer, I will be pleased if you can confirme your assertion. – Yassine Guerboussa Jul 01 '13 at 12:37
Which one...?... – DonAntonio Jul 01 '13 at 12:55
but I think in general the answer is negative since there are simple (infinite) p−groups – Yassine Guerboussa Jul 01 '13 at 13:06
Oh, that: google "Tarski Monsters"...:) – DonAntonio Jul 01 '13 at 13:09
Ok, I know that a Tarski monster is simple, I would like to know if this implies that is not engel. – Yassine Guerboussa Jul 01 '13 at 13:18

Are $p$-groups Engel groups?

2 Answers2