Difference between revisions of "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==== | ====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
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