View source for Binary p-adic-group

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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| $  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.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1979)  pp. 482–490</TD></TR></table>