Homomorphic images of Coxeter groups in the Golod \(2\)-group

A. Timofeenko

A Coxeter group is a group generated by a finite number of involutions whose set of defining relations consists only of the degree of the product every two generators.

In 1959, P. S. Novikov announced the existence of finitely generated infinite groups of bounded exponent. While preparing a proof of this result, E. S. Golod built for every prime \(p\) and every \(n \ge 2\), an infinite \(n\)-generated \(p\)-group [1]. Recall that the basis for the construction of this group is a sufficient condition for the infinite dimensionality of the quotient algebra of the free associative algebra \(F^{(1)}\) of polynomials without constant term in \(n\) non-commuting variables over an arbitrary field by a homogeneous ideal \(I.\) This condition limits the number of polynomials of each degree in the generating set of \(I\). Each polynomial of \(F^{(1)}\) is raised to a sufficiently high power so that the homogeneous components of this power are taken as generators of \(I\) and the number of these generators of each degree satisfies the above-mentioned condition for infinite dimensionality. In this way, a nonnilpotent nilalgebra \(A = F^{(1)}/I\) is built. The algebra \(A\) over a field of characteristic \(p\) under the operation \(\circ:\) \[ a \circ b = a + b + ab, \, \, \, a, b \in A, \] forms an infinite \(p\)-group, which is called the adjoint group of \(A.\) We denote by \(x_1, x_2, \ldots, x_n\) the free generators of \(F^{(1)}\), and let \(a_j = x_j + I, \, j = 1,2, \ldots, n.\) Then the subgroup \(\langle a_1, a_2, \ldots, a_n \rangle\) of the adjoint group of \(A\) will also be infinite. We call it the Golod group.

The Golod group is residually finite and its element orders are not bounded. There are Golod groups with infinite center [2] and with trivial center [3,4]. As shown in [5], the ideal \(I\) can be constructed so that each \((n-1)\)-generated subgroup of the Golod group is finite. On the other hand, for every \(n = 2,3, \ldots\) and every odd prime \(q\), one can construct an \(n\)-generated Golod \(q\)-group with an infinite subgroup generated by two conjugate elements of prime order [6,7]. A Golod 2-group with an infinite subgroup generated by a pair of conjugate elements of order four is constructed in [7]. These subgroups can be constructed so as to satisfy the sufficient condition for the infiniteness of index in [8]. The subgroups in the following theorem can have same properties.

Theorem. For every \(n \geq 3\) and every \(m \geq 3\), the \(n\)-generated Golod \(2\)-group with an infinite subgroup generated by \(m\) involutions is constructed.

References

1. E. S. Golod, On nil-algebras and finitely approximable \(p\)-groups, Izv. Akad. Nauk SSSR, Ser. Mat., 28, N2 (1964), 273–276. (Russian)
2. A. V. Timofeenko, The existence of Golod groups with infinite center, Math. Notes, 39, N5 (1986), 353–355.
3. V. A. Sereda, A. I. Sozutov, On a question in the Kourovka notebook, Math. Notes, 80, N1 (2006), 151–153.
4. V. A. Sereda, A. I. Sozutov, Associative nil-algebras and Golod groups, Algebra Logic, 45, N2 (2006), 134–138.
5. E. S. Golod, Some problems of Burnside type, Proc. Internat. Math. Congr. (Moscow, 1966), Mir, Moscow (1968), 284–289. (Russian)
6. A. V. Timofeenko, \(2\)-generator Golod \(p\)-groups, Algebra Logic, 24, N2 (1985), 129–139.
7. A. V. Timofeenko, Subgroups of periodic non-locally finite groups of Golod and Aleshin types, Ph.D. Phys.-Math. Thesis, Krasnoyarsk (1991). (Russian)
8. A. V. Timofeenko, Infinite subgroups of infinite index in \(2\)-generated \(p\)-groups of Golod type, Sib. Mat. Zh., 27, N5(159) (1986) 194–195. (Russian)

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