“Verbal subgroup”的意思、由来-开放百科全书

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Verbal subgroup

释义

References

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 xy, the corresponding verbal subgroup is 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 is generated by the set of squares and their conjugates. Verbal subgroups are the only fully characteristic subgroups of a free group and therefore represent the generic example of fully characteristic subgroups, {{harv|Magnus|Karrass|Solitar|2004|p=75}}.

Another example is the verbal subgroup of , which is the derived subgroup.

References

{{Citation | last1=Magnus | first1=Wilhelm | author1-link=Wilhelm Magnus | last2=Karrass | first2=Abraham | first3=Donald | last3=Solitar | title=Combinatorial Group Theory | publisher=Dover Publications | location=New York | isbn=978-0-486-43830-6 | mr=0207802 | year=2004}}

2 : Infinite group theory|Subgroup properties

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