Proof: Basic Properties of Homomorphisms (Identities and Inverses) | Abstract Algebra

We prove two basic properties of a homomorphism f from a group G to a group H. First, we prove the homomorphism maps the identity of G to the identity of H, that is: f(e) = e. Then we prove the homomorphism maps inverses of a to corresponding inverses in H. So, f(a^-1) = [f(a)]^-1. #abstractalgebra #grouptheory
Group Homomorphisms: • Intro to Group Homomor...
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[@OmarWehbeh-hXO]

Why is f^-1(x)*f(x) = e_H?

[@WrathofMath]

f(x) and its inverse f^(-1)(x) both exist in H, the codomain of the homomorphism because they are images of the function. Since they are inverses, they undo each other and thus produce the identity of the appropriate group, which is H. Does that help?

[@OmarWehbeh-hXO]

@@WrathofMath This really explained everything, Thank you so much🙏 You're the best👊👍