Generators of a free groupIf G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?

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Dillan Foley · Accepted Answer

Here is another approach. Let   G be a free group on   m generators, and let   H be a free group on   n generators. There are exactly       2    m   homomorphisms from   G to a group of order two, since each generator can be mapped in two ways. Likewise, there are       2    n   homomorphisms from   H to a group of order two.If   G and   H are isomorphic, then they have the same number of homomorphisms to a group of order two. Therefore       2    m    =      2    n  , which implies   m  =  n

Finn Mosley · Accepted Answer

No, if       F    n   and       F    m   are isomorphic, then their abelianizations are also isomorphic, which only happens when n=m (for example, reduce mod p and count points).

Generators of a free group If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?

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