In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements for variables in a given set of words. For example, given the word , the corresponding verbal subgroup is the group generated by all squares.
Examples For a given group the following are examples of verbal subgroups: The entire group , given by the word . The trivial subgroup, given by the empty word. The commutator subgroup, given by the word . Furthermore every subgroup in the derived series is a verbal subgroup. The th term of the derived series is given by the set of words where is a free group with at least as many generators as . Every subgroup in the lower central series is a verbal subgroup. The th term of the lower central series is given by the set of words where is a free group with at least as many generators as .
Properties
Verbal subgroups are fully characteristic subgroups. That is they are closed under endomorphisms of the ambient group. Since all fully characteristic subgroups are normal subgroups, verbal subgroups are normal. In free groups, the verbal subgroups are exactly the fully characteristic subgroups. Therefore verbal subgroups represent the generic example of fully characteristic subgroups. The lattice of verbal subgroups of the countably generated free group under inclusion is anti-isomomorphic to the lattice of group varieties under inclusion.
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