Question

Question: If $f:G\to H$is an isomorphism of groups and if T is a subgroup of G , prove that is T isomorphic to the subgroup$f\left(T\right)=\left\{f\left(a\right)\right|a\in T\}$ of H .

Short answer

It has been proved that T is isomorphic to the subgroup $f\left(T\right)=\left\{f\left(a\right)\right|a\in T\}$of H .

Step-by-step solution

Step-By-Step SolutionStep 1: Define a restrictive map

Let , $f_1:T\to H$is a restricted map of f.

It is still an injective homomorphism.

Use Theorem 7.20 to prove isomorphism

Theorem 7.20 states that, “Let G and H be groups such that $f:G\to H$is an injective homomorphism, then $G\cong Imf$”

Since$f_1$is also an injective homomorphism, therefore,

$f\left(T\right)=f_1\left(T\right)=Im{f}_1$

Thus $f_1$induces an isomorphism.

Conclusion

Thus,$T\cong f\left(T\right)$$T\cong f\left(T\right)$