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verbal subgroup
In mathematics, especially in the area of abstract algebra known as group theory, a verbal subgroup is any subgroup of a group definable as the subgroup generated by the set of all elements formed by choices of elements for a given set of words. For example, given the word ''xy'', the corresponding verbal subgroup of would be generated by the set of all products of two elements in the group, substituting any element for ''x'' and any element for ''y'', and hence would be the group itself. On the other hand the verbal subgroup of would be generated by the set of squares and their conjugates. Verbal subgroups are particularly important as the only fully characteristic subgroups of a free group and therefore represent the generic example of fully characteristic subgroups, .
Another example is the verbal subgroup of , which is the derived subgroup.
== References ==
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抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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