Is it possible for a finitely-generated group to contain subgroups that are not finitely-generated ? True or False. Prove your conclusion.
It is possible.
The classic example is the commutator subgroup of the free group on two generators.
#G = < a, b>#
#[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
Strip out any element-inverse element pairs to reduce the product to minimum form, i.e. get rid of combinations like
#a a^(-1)#or #b^(-1) b#.
The cleaned up form will start with a block of
#a^(-1)#'s or #b^(-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.
Repeat with the remainder of the product to recover the remaining commutators.
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