Verbality is transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., verbal subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Suppose H \le K \le G are groups with each verbal in the next (i.e., G is a group, K is a verbal subgroup of G, and H is a verbal subgroup of K). Then, H is a verbal subgroup of G.

Related facts


Proof idea

The idea is to "compose" the words by substituting. Explicitly, any element of H can be written as a word of a certain type in terms of elements of K, and each of those elements of K can be written as words of certain types in the element of G. We plug in those word expressions. Explicitly, if:

h = w(k_1,k_2,\dots,k_n)


k_i = w_i(\mbox{elements of } G)


h = (w \circ (w_1,w_2,\dots,w_n))(\mbox{elements of } G)