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An infinite group $G$ of square matrices of order 2

$$\left \| \begin{array}{lr} a & b \\ c & d \\ \end{array} \right \| ,$$

where $a, b, c, d$ are integral $p$- adic numbers (cf. $p$- adic number) satisfying the following conditions:

$$ad - bc = 1,\ \ c \equiv 0 ( \mathop{\rm mod} p),\ \ d \equiv 1 ( \mathop{\rm mod} p).$$

The quotient groups of such groups of the form $G/N$, where $N$ is the $n$- th member of the lower central series of $G$ or the $n$- th term of the derived series (the series of higher commutators of $G$), are examples of finite $p$- groups having certain extremal properties.