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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<strong>F<strong>IN</strong>D<strong>IN</strong>G</strong> N-<strong>TH</strong> <strong>ROOTS</strong> <strong>IN</strong> <strong>NILPOTENT</strong> <strong>GROUPS</strong>... 585<br />
6. The Heisenberg Group<br />
6.1. Roots in H3(Z)<br />
Recall that the Heisenberg group is nilpotent of class 2 and that any element<br />
can be written in the form D = A m B n C k , where<br />
⎛ ⎞<br />
1 1 0<br />
⎛ ⎞<br />
1 0 0<br />
⎛ ⎞<br />
1 0 1<br />
A = ⎝0<br />
1 0⎠,<br />
B = ⎝0<br />
1 1⎠<br />
and C = ⎝0<br />
1 0⎠<br />
.<br />
0 0 1 0 0 1<br />
0 0 1<br />
That is, we can write<br />
D = A m B n C k =<br />
⎛ ⎞<br />
1 m k + mn<br />
⎝0<br />
1 n ⎠.<br />
0 0 1<br />
⎛ ⎞<br />
1 a c<br />
Proposition 6.1. Let r ∈ N. The matrix M = ⎝0<br />
1 b⎠<br />
= A<br />
0 0 1<br />
aBbC c−ab<br />
has an r-th root in the Heisenberg group if and only if a ≡ 0 (mod r), b ≡ 0<br />
+ 1) ≡ 0 (mod r).<br />
(mod r) and c − 1<br />
2ab(1 r<br />
Proof. Suppose that<br />
This shows that<br />
M = D r = (A m B n C k ) r =<br />
a = mr<br />
⎛<br />
1 mr rk +<br />
⎝<br />
1 ⎞<br />
2rmn(1 + r)<br />
0 1 nr ⎠<br />
0 0 1<br />
= A mr B nr C rk+1<br />
2 rmn(1−r)<br />
b = nr<br />
c − ab = rk + 1<br />
rmn(1 − r)<br />
2<br />
Hence, a ≡ 0 (mod r), b ≡ 0 (mod r), and c − ab = rk + 1<br />
2<br />
rk + ab<br />
2 (1<br />
r<br />
− 1), or c − 1<br />
2<br />
1 ab(1 + r ) ≡ 0 (mod r).<br />
r · a<br />
r<br />
b · r (1 − r) =