FINDING N-TH ROOTS IN NILPOTENT GROUPS AND ...
FINDING N-TH ROOTS IN NILPOTENT GROUPS AND ...
FINDING N-TH ROOTS IN NILPOTENT GROUPS AND ...
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572 S. Sze, D. Kahrobaei, R. Dambreville, M. Dupas<br />
the center of H. Equivalently, [Ai+1,G] ≤ Ai.<br />
Definition 1.2. The lower central series for a group G is a series of<br />
subgroups<br />
G = G0 G1 · · · Gk · · ·<br />
where Gk+1 = [Gk,G] for k ≥ 0.<br />
Definition 1.3. A group G is said to be nilpotent if it has a lower central<br />
series of finite length.<br />
Definition 1.4. The upper central series for a group G is a sequence of<br />
subgroups<br />
{1} = Z0 Z1 · · · Zk · · ·<br />
such that Z1 = Z(G) and Zk+1/Zk = Z(G/Zk).<br />
In any nilpotent group the upper and lower central series terminates at the<br />
same n.<br />
Definition 1.5. The class of a nilpotent group G is the smallest such<br />
n such that the central series terminates. Equivalently, it is the length of the<br />
lower (or upper) central series for the nilpotent group G. We then say that G<br />
is nilpotent of class n.<br />
Clearly, any abelian group is nilpotent of class 1.<br />
Let G be nilpotent of class 2. This means G has the following lower central<br />
series:<br />
G [G,G] [[G,G],G] = {1}.<br />
We have the following results for G:<br />
• [G,G] ≤ Z(G): this is clear since [[G,G],G] = {1}.<br />
• For all x,y,z ∈ G, [x,yz] = [x,y][x,z]: this is true since [x,y][x,z] =<br />
x −1 y −1 xyx −1 z −1 xz = x −1 (y −1 xyx −1 )z −1 xz = x −1 z −1 (y −1 xyx −1 )xz =<br />
x −1 z −1 y −1 xyz = [x,yz].<br />
• For all x,y,z ∈ G, [xy,z] = [x,z][y,z].<br />
• For all x,y ∈ G, [x m ,y n ] = [x,y] mn .<br />
Definition 1.6. The Heisenberg group, H3(R), is an infinite group consisting<br />
of upper triangular 3 × 3 matrices of the form<br />
⎛ ⎞<br />
1 a c<br />
⎝0<br />
1 b⎠,<br />
0 0 1