can someone help me with this questuon please its fundamnetal to the topic and preventing me understand the topic!!
Thanks
Edgar
if G is a group and x is an element of G we define the order ord(x) of x by
ord(x) = min {r > or equal to 1 : x^r=1}
if f:G maps to His an injective group homomorphism show that for each x
ord (f(x)) = ord (x)
Because, is one-to-one.
Thus, a injective function between two finite sets.
Now, thus, (again they are finite).
But you cannot have a injective map from where they are non-empty and . Thus, the only possible choice is that , that is the subgroup is the group itself. Now a injective map between the same finite non-empty set is also surjective (Pigeonhole principle). Thus, what we really have and it is an automorphism.
Thus, by homomorphism properties,
*
(The reason why is because it is an automorphism).
Thus, we know the order of is at most . If it was less than by * we have that has smaller order which cannot be contrary to our assumption.