Nilpotent semidirect products
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From: Lilly Bougainville on 22 Feb 2010 13:25 Let G be a the semidirect product (K,Q,f), with Q acting on K via f. When is G nilpotent? It is clear that K and Q must be nilpotent, so it really depends on the action. By the way, I'm more interested in infinite groups, but anything (specially examples) is welcome. Thanks a lot... Lilly
From: Derek Holt on 23 Feb 2010 04:55 On 22 Feb, 18:25, Lilly Bougainville <lbougainvi...(a)gmail.com> wrote: > Let G be a the semidirect product (K,Q,f), with Q acting on K via f. > When is G nilpotent? > It is clear that K and Q must be nilpotent, so it really depends on > the action. > By the way, I'm more interested in infinite groups, but anything > (specially examples) is welcome. > Well yes, G is nilpotent if and only if the action is nilpotent i.e. if and only if the left-normed commutator subgroup [K,Q,Q,...Q] = 1 for some finite number of repetitions of Q. The same would be true for any extension G of K by Q, not just the semidirect product. I am not sure if that helps you much! As an example (with K abelian), let F be a field, K=(F^n,+) for some n>1, and let Q be the group of upper or lower unitriangular nxn matrices with entries in F, with the action given by the natural action of the matrices on F^n. The semidirect product is a nilpotent group of class n. Derek Holt.
From: Lilly Bougainville on 25 Feb 2010 04:27
On Feb 23, 10:55 am, Derek Holt <ma...(a)warwick.ac.uk> wrote: > On 22 Feb, 18:25, Lilly Bougainville <lbougainvi...(a)gmail.com> wrote: > > > Let G be a the semidirect product (K,Q,f), with Q acting on K via f. > > When is Gnilpotent? > > It is clear that K and Q must benilpotent, so it really depends on > > theaction. > > By the way, I'm more interested in infinite groups, but anything > > (specially examples) is welcome. > > Well yes, G isnilpotentif and only if theactionisnilpotenti.e. > if and only if the left-normed commutator subgroup > > [K,Q,Q,...Q] = 1 > > for some finite number of repetitions of Q. The same would be true for > any extension G of K by Q, not just the semidirect product. > > I am not sure if that helps you much! > > As an example (with K abelian), let F be a field, K=(F^n,+) for some > n>1, and let Q be the group of upper or lower unitriangular nxn > matrices with entries in F, with theactiongiven by the naturalactionof the matrices on F^n. The semidirect product is anilpotent > group of class n. > > Derek Holt. Could you give me any reference for this definition? thank you so much... Lilly |