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Difference between revisions of "Binary p-adic-group"

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An infinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163602.png" /> of square matrices of order 2
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163603.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163604.png" /> are integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163605.png" />-adic numbers (cf. [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163606.png" />-adic number]]) satisfying the following conditions:
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An infinite group  $  G $
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of square matrices of order 2
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163607.png" /></td> </tr></table>
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$$
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\left \|
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 +
\begin{array}{lr}
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a  & b \\
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c  & d  \\
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\end{array}
  
The quotient groups of such groups of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163608.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b0163609.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b01636010.png" />-th member of the lower central series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b01636011.png" /> or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b01636012.png" />-th term of the derived series (the series of higher commutators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b01636013.png" />), are examples of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016360/b01636014.png" />-groups having certain extremal properties.
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\right \| ,
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$$
  
 +
where  $  a, b, c, d $
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are integral  $  p $-
 +
adic numbers (cf. [[P-adic number| $  p $-
 +
adic number]]) satisfying the following conditions:
  
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$$
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ad - bc  =  1,\ \
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c  \equiv  0  (  \mathop{\rm mod}  p),\ \
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d  \equiv  1  (  \mathop{\rm mod}  p).
 +
$$
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The quotient groups of such groups of the form  $  G/N $,
 +
where  $  N $
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is the  $  n $-
 +
th member of the lower central series of  $  G $
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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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1979)  pp. 482–490</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1979)  pp. 482–490</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


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.

Comments

References

[a1] B. Huppert, "Endliche Gruppen" , 1 , Springer (1979) pp. 482–490
How to Cite This Entry:
Binary p-adic-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_p-adic-group&oldid=46063
This article was adapted from an original article by A.I. Kostrikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article