# Verbality is transitive

From Groupprops

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)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about verbal subgroup |Get facts that use property satisfaction of verbal subgroup | Get facts that use property satisfaction of verbal subgroup|Get more facts about transitive subgroup property

## Contents

## Statement

Suppose are groups with each verbal in the next (i.e., is a group, is a verbal subgroup of , and is a verbal subgroup of ). Then, is a verbal subgroup of .

## Related facts

- Verbality is quotient-transitive
- Verbality is strongly join-closed
- Verbality satisfies image condition

## Proof

### Proof idea

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

where:

then: