Verbal subgroup

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 ${\displaystyle x^{2}}$, the corresponding verbal subgroup is the group generated by all squares.

Examples

For a given group ${\displaystyle G}$ the following are examples of verbal subgroups:

• The entire group ${\displaystyle G}$, given by the word ${\displaystyle x}$.
• The trivial subgroup, given by the empty word.
• The commutator subgroup, given by the word ${\displaystyle [x,y]=xyx^{-1}y^{-1}}$.
  • Furthermore every subgroup in the derived series is a verbal subgroup. The ${\displaystyle n}$th term of the derived series ${\displaystyle G^{(n)}}$ is given by the set of words ${\displaystyle \{[w,v]\mid w,v\in F^{(n-1)}\}}$ where ${\displaystyle F}$ is a free group with at least as many generators as ${\displaystyle G}$.
  • Every subgroup in the lower central series is a verbal subgroup. The ${\displaystyle n}$th term of the lower central series ${\displaystyle G_{n}}$ is given by the set of words ${\displaystyle \{[w,x]\mid w\in F_{n-1}\}}$ where ${\displaystyle F}$ is a free group with at least as many generators as ${\displaystyle G}$.

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.

References

Bibliography

• Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004), Combinatorial Group Theory, New York: Dover Publications, ISBN 978-0-486-43830-6, MR 0207802
• Neumann, B. H. (1967), Varieties of groups (PDF), pp. 603–613, retrieved 2026-06-28

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