FINDING N-TH ROOTS IN NILPOTENT GROUPS AND ...

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586 S. Sze, D. Kahrobaei, R. Dambreville, M. Dupas ⎛ ⎞ 1 a c Corollary 6.2. A matrix D = ⎝0 1 b⎠ in the Heisenberg group has 0 0 1 a square root if and only if a ≡ 0 (mod 2), b ≡ 0 (mod 2) and c − 3ab 4 ≡ 0 (mod 2). In the proof of Proposition 6.1, we showed how to find the r-th root of a matrix in H3(Z). In particular, if we want to find the square root of M = A a B b C c−ab , we only need to solve the system of equations a = 2m b = 2n c − ab = 2k − mn, for m, n, and k and we will see that the square root of M is D = A m B n C k . Example 6.3. Given that D2 ⎛ ⎞ 1 4 26 = ⎝0 1 6 ⎠ in the Heisenberg group, 0 0 find D. 1 We have a = 4 and b = 6. This means that m = 2 and n = 3. Next, 26−4(6) = 2k−(2)(3) implies that k = 4. Hence, D = A2B3C 4 ⎛ ⎞ 1 2 10 = ⎝0 1 3 ⎠. 0 0 ⎛ ⎞ 1 −12 5 1 Example 6.4. Does ⎝0 1 6⎠ have a square root in the Heisenberg 0 group? 0 1 We have a = −12, b = 6 and c = 5. Clearly, a ≡ b ≡ 0 (mod 2). However, c − 3ab 3(−12)(6) 4 = 5 − 4 = 5 + 54 ≡ 0 (mod 2). Therefore, the given matrix does not have a square root in the Heisenberg group. 7. Engel Groups Definition 7.1. A group is said to be n−Engel if for every pair (x,y) ∈ G × G there exists an n = n(x,y) such that [· · · [[x,y],y] · · · y] = 1. n Here are some theorems related to n-Engel groups and nilpotent groups (see [17] for more information).
FINDING N-TH ROOTS IN NILPOTENT GROUPS... 587 Theorem 7.2. (Zorn, 1936) Every finite Engel group is nilpotent. Theorem 7.3. (Gruenberg, 1953) Every finitely generated solvable Engel group is nilpotent. Theorem 7.4. (Baer, 1957) Every Engel group satisfying max is nilpotent. Theorem 7.5. (Suprenenko and Garscuk, 1962) Every linear Engel group is nilpotent. Proposition 7.6. The Heisenberg group is a 2-Engel group. Proof. We want to show that [[X,Y ],Y ] = I. Let X = A m B n C k = ⎛ ⎞ 1 m k + mn ⎝0 1 n ⎠ and Y = A 0 0 1 uBvC w ⎛ ⎞ 1 u w + uv = ⎝0 1 v ⎠. Then X 0 0 1 −1 = ⎛ ⎞ 1 −m −k ⎝0 1 −n⎠ and Y 0 0 1 −1 ⎛ ⎞ 1 −u −w = ⎝0 1 −v⎠. Next, [X,Y ] = X 0 0 1 −1Y −1XY = ⎛ ⎞ 1 0 −nu + mv ⎝0 1 0 ⎠. Finally, [[X,Y ],Y ] = [X,Y ] 0 0 1 −1Y −1 ⎛ ⎞ 1 0 0 [X,Y ]Y = ⎝0 1 0⎠. 0 0 1 Lemma 7.7. Every element in the Heisenberg group satisfies Y X 2 Y = XY 2 X. Proof. We will prove a more general case: in every 2-Engel group, yx 2 y = xy 2 x. To see this, note that [y,x][x −1 ,y −1 ] = [y,x]xyx −1 y −1 = x[y,x]yx −1 y −1 = x[y,x]yx −1 y −1 xx −1 = x[y,x][y −1 ,x]x −1 = x[e,x]x −1 = e. From this, we conclude that [y,x] = [x −1 ,y −1 ] −1 = [y −1 ,x −1 ], which is equivalent to yx 2 y = xy 2 x. There are other semigroup laws for other Engel groups. Here, we will discuss the laws for 3-Engel and 4-Engel groups.
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