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In this note we give simple explicit examples of free subgroups of rank 2 in the group of infinite upper unitriangular matrices over integers. The proofs that the given subgroup is free are elementary. Using canonical projections mod p (p — any odd prime) we obtain also free subgroups in the group of finite state automata on alphabet of size p, extending results of Aleshin [1], Brunner–Sidki [2] and Olijnyk–Sushchansky [8, 9] in this direction. As application, we give the simple proofs of two classical results on free groups.

A binary word wxY y is called 2-symmetric for a given group G if wgY h whY g for all gY h in G. In this note we describe 2-symmetric words for the relatively free (nilpotent of class 2)-by-abelian groups and for the relatively free centreby-metabelian groups.1. Introduction. Symmetric words for a group G are closely related to the fixed points of the automorphisms permuting generators in their corresponding relatively free groups. The problem of characterizing the 2-symmetric words for a given group G was initiated by Pøonka ([6], [7]) who, among other things, gave a complete description of the 2-symmetric words for nilpotent groups of class % 3. Further descriptions of 2-symmetric words are now known for the free metabelian groups and free soluble groups of derived length 3 (Macedon  ska and Solitar [4]) and free metabelian nilpotent groups of arbitrary nilpotency class c (Hoøubowski [2]). Further work for free nilpotent groups is under investigation by the second author ([3]). In this note we extend the work in [4] and characterize the 2-symmetric words for free centre-by-metabelian groups (Theorem 3) and for free (nilpotent of class 2)-by-abelian groups (Theorem 2).