Binary p-adic-group

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.

References

[a1]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1979) pp. 482–490