Properties of Isomorphism

Theorem 1:
If isomorphism exists between two groups, then the identities corresponds, i.e. if f:G \to G' is an isomorphism and e,e' are respectively the identities in G,\,G', then f\left( e \right) = e'.

Theorem 2:
If isomorphism exists between two groups, then the identities corresponds, i.e. if f:G \to G' is an isomorphism and f\left( a \right) = a', where a \in G,\,\,\,a' \in G' then f\left( {{a^{ - 1}}} \right) = {a'^{ - 1}} =  {\left[ {f\left( a \right)} \right]^{ - 1}}.

Theorem 3:
In an isomorphism the order of an element is preserved, i.e. if f:G  \to G' is an isomorphism, and the order of a is n, then the order of f\left( a \right) is also n.
Proof: As f\left( a \right) = a', then we have f\left( {a \cdot a} \right) = f\left( a  \right) \cdot f\left( a \right) = a' \cdot a' = {a'^2} and in general we can write it as f\left( {{a^n}} \right) =  {a'^n}.
But f\left( {{a^n}} \right) = f\left( e \right) = e', by using the statement of theorem 1
Therefore {a'^n} = e'. Also {a'^m} \ne e' for m < n, i.e. o\left( {a'} \right) = n.
If follows that the order of an element of G, if finite, is equal to the order of its image in G'. If the order of a is infinite, we can similarly show that the order of a' can not be finite.

Theorem 4:
The relation of isomorphism in the set of groups is an equivalence relation.

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