Monomial substitutions, group of

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)