Is it possible for a finitely-generated group to contain subgroups that are not finitely-generated ? True or False. Prove your conclusion.
1 Answer
It is possible.
Explanation:
The classic example is the commutator subgroup of the free group on two generators.
Let:
G = < a, b>
If
[g, h] = g^(-1) h^(-1) g h
The subgroup of
We can make things simpler by specifying the subgroup
Any element of
Given a product of
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Strip out any element-inverse element pairs to reduce the product to minimum form, i.e. get rid of combinations like
a a^(-1) orb^(-1) b . -
The cleaned up form will start with a block of
a^(-1) 's orb^(-1) 's. The length of this block allows you to identify the first commutator in a representation as a product of commutators of the form[a^n, b^n] and/or[b^n, a^n] . -
The end of the commutator may require some element-inverse element pairs to be added in order to reconstitute it. Add just as many as necessary before splitting off the first commutator.
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Repeat with the remainder of the product to recover the remaining commutators.
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Once all commutators have been recovered, strip out any adjacent pairs of commutators of the form
[a^n, b^n][b^n, a^n] or[b^n, a^n][a^n b^n] to reduce to the minimum representation.
Hence we find that