Hi I am new to Group Theory and I am confused by a problem :
Let be any group of order and be an integer prime to .
Is it true that ?
I know there exists such that
Then
I think if , then i can prove it but is it really true ? Or my proof has some mistakes ? Actually , it is what i guess and i don't know if it is correct .
Thanks !
The property is called the `unique roots property', and it does not always hold. It does hold in, for example, free groups (infinite groups that can be viewed as the sort of top-down building blocks of group theory).
Such groups have the property that .
However, this property does not always hold. A finite example would be the elements and . Then but .
What I believe you proved in your post is that if and then if we have that .
(An infinite example of a group without the unique roots property is the group given by the presentation (rules) . This is non-trivial and infinite as it has a homomorphism onto the infinite dihedral group, )