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The subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647901.png" /> of all invertible matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647902.png" /> over the integral group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647903.png" /> (see [[Group algebra|Group algebra]]) of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647904.png" />, consisting of all matrices which precisely contain one non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647905.png" /> in each row and column. Each such matrix, having a non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647906.png" /> in place <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647907.png" />, corresponds to a monomial substitution, that is, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647908.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m0647909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479011.png" /> is a permutation of the finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479012.png" />. The product of such mappings is given by the formula
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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/m/m064/m064790/m06479013.png" /></td> </tr></table>
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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479015.png" />), and corresponds to the product of the matrices associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479017.png" />. Any group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479018.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479019.png" /> as a subgroup of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479020.png" /> can be isomorphically imbedded in a group of monomial substitutions. The group of monomial substitutions is isomorphic to the (unrestricted) [[Wreath product|wreath product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479021.png" /> with the [[Symmetric group|symmetric group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479022.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064790/m06479023.png" />.
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The subgroup of the group  $  \mathop{\rm GL} ( m , \mathbf Z [ H ] ) $
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of all invertible matrices of order  $  m $
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over the integral group ring  $  \mathbf Z [ H ] $(
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see [[Group algebra|Group algebra]]) of a group  $  H $,
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consisting of all matrices which precisely contain one non-zero element of  $  H $
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in each row and column. Each such matrix, having a non-zero element  $  h _ {ij} \in H $
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in place  $  ( i , j ) $,
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corresponds to a monomial substitution, that is, a mapping  $  \psi :  u _ {i} \rightarrow h _ {ij} u _ {j} $,  
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where  $  j = j ( i ) $,
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$  i = 1 \dots m $,
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and  $  u _ {i} \rightarrow u _ {j} $
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is a permutation of the finite set  $  S = \{ u _ {1} \dots u _ {m} \} $.  
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The product of such mappings is given by the formula
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$$
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\psi _ {1} \psi _ {2} : \
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u _ {i}  \rightarrow  ( h _ {ij} h _ {ik} ) u _ {k}  $$
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( $  \psi _ {1} :  u _ {i} \rightarrow h _ {ij} u _ {j} $,
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$  \psi _ {2} :  u _ {j} \rightarrow h _ {ik} u _ {k} $),  
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and corresponds to the product of the matrices associated with $  \psi _ {1} $
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and $  \psi _ {2} $.  
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Any group $  G $
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containing $  H $
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as a subgroup of index m $
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can be isomorphically imbedded in a group of monomial substitutions. The group of monomial substitutions is isomorphic to the (unrestricted) [[Wreath product|wreath product]] of $  H $
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with the [[Symmetric group|symmetric group]] $  S ( m ) $
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of degree m $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


The subgroup of the group $ \mathop{\rm GL} ( m , \mathbf Z [ H ] ) $ of all invertible matrices of order $ m $ over the integral group ring $ \mathbf Z [ H ] $( see Group algebra) of a group $ H $, consisting of all matrices which precisely contain one non-zero element of $ H $ in each row and column. Each such matrix, having a non-zero element $ h _ {ij} \in H $ in place $ ( i , j ) $, corresponds to a monomial substitution, that is, a mapping $ \psi : u _ {i} \rightarrow h _ {ij} u _ {j} $, where $ j = j ( i ) $, $ i = 1 \dots m $, and $ u _ {i} \rightarrow u _ {j} $ is a permutation of the finite set $ S = \{ u _ {1} \dots u _ {m} \} $. The product of such mappings is given by the formula

$$ \psi _ {1} \psi _ {2} : \ u _ {i} \rightarrow ( h _ {ij} h _ {ik} ) u _ {k} $$

( $ \psi _ {1} : u _ {i} \rightarrow h _ {ij} u _ {j} $, $ \psi _ {2} : u _ {j} \rightarrow h _ {ik} u _ {k} $), and corresponds to the product of the matrices associated with $ \psi _ {1} $ and $ \psi _ {2} $. Any group $ G $ containing $ H $ as a subgroup of index $ m $ can be isomorphically imbedded in a group of monomial substitutions. The group of monomial substitutions is isomorphic to the (unrestricted) wreath product of $ H $ with the symmetric group $ S ( m ) $ of degree $ m $.

References

[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)
[3] M. Hall jr., "The theory of groups" , Macmillan (1959)
How to Cite This Entry:
Monomial substitutions, group of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monomial_substitutions,_group_of&oldid=47892
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article