3-local lower central series: Difference between revisions

Revision as of 15:00, 27 June 2013

Definition

The 3-local lower central series of a group $G$ is a descending series defined as follows. The $i^{t h}$ member, which we will denote as $γ_{i}^{3 - l o c} (G)$ is defined as:

$γ_{i}^{3 - l o c} (G) = ⟨ γ_{i} (H) ⟩$ where $H$ varies over all subgroups of $G$ that are generated by at most 3 elements and $γ_{i} (H)$ denotes the $i^{t h}$ member of the lower central series of $H$ .

$G$ has 3-local nilpotency class (at most) $c$ if and only if $γ_{c + 1}^{3 - l o c} (G)$ is the trivial subgroup.

Revision as of 14:36, 27 June 2013 (view source) Vipul (talk \| contribs) (Created page with "==Definition== The '''3-local lower central series''' of a group <math>G</math> is a descending series defined as follows. The <math>i^{th}</math> member, which we will d...")		Revision as of 15:00, 27 June 2013 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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	<math>\gamma_i^{3-loc}(G) = \langle \gamma_i(H) \rangle</math> where <math>H</math> varies over all subgroups of <math>G</math> that are generated by at most 3 elements and <math>\gamma_i(H)</math> denotes the <math>i^{th}</math> member of the [[lower central series]] of <math>H</matH>.		<math>\gamma_i^{3-loc}(G) = \langle \gamma_i(H) \rangle</math> where <math>H</math> varies over all subgroups of <math>G</math> that are generated by at most 3 elements and <math>\gamma_i(H)</math> denotes the <math>i^{th}</math> member of the [[lower central series]] of <math>H</matH>.

	~~<math>G</math> is a [[3-locally nilpotent group]] if and only if its 3-local lower central series reaches the trivial subgroup in finitely many steps. Further,~~ <math>G</math> has [[3-local nilpotency class]] (at most) <math>c</math> if and only if <math>\gamma_{c + 1}^{3-loc}(G)</math> is the trivial subgroup.		<math>G</math> has [[3-local nilpotency class]] (at most) <math>c</math> if and only if <math>\gamma_{c + 1}^{3-loc}(G)</math> is the trivial subgroup.