Verbal implies fully invariant
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., verbal subgroup) must also satisfy the second subgroup property (i.e., fully invariant subgroup)
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|verbal subgroup||A subgroup of is a verbal subgroup if there exists a collection of words such that is the union of the images of the word maps corresponding to all the words in .|
|fully invariant subgroup||A subgroup of is a fully invariant subgroup if for all endomorphisms of .|
For reduced free groups
- Fully invariant implies verbal in reduced free group: For a reduced free group, a verbal subgroup is the same thing as a fully characteristic subgroup. The set of words corresponding to a given fully characteristic subgroup is simply the set of words in the corresponding free generators.
- Homomorphism commutes with word maps
- Image of invariant set for a map under a commuting map is also invariant for the map
Given: A group and subgroup . is a collection of words such that an element is in if and only if it can be expressed, as the value of a word in with letters taken as elements of . In other words, is the union of the images of the word maps corresponding to the elements of . (Note that the existence of such a is what makes a verbal subgroup). An endomorphism of and element .
To prove: .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||There exists and elements (where is the number of letters in ) such that .||From given condition on ( being verbal) and .|
|2||Fact (1)||is an endomorphism of .||Step (1)||The second equality is direct from Fact (1), noting that an endomorphism of just means a homomorphism from to itself.|
|3||is in the image of the word map corresponding to , hence .||Given condition on .||Steps (1), (2)||Step (2) shows that is in the image of the word map corresponding to , with inputs . Step (1) reminds us that , so by the definition of .|
Note that any endomorphism is a homomorphism, so every endomomorphism of commutes with every word map of by Fact (1). Fact (2) now tells us that the image of every word map is invariant under every endomorphism of . Since a union of invariant subsets under a bunch of functions is still invariant, we obtain that any union of images of word maps is invariant under every endomorphism. In particular, any verbal subgroup is invariant under every endomorphism.