group theory

[YYDS]

Let φ : G → H be a homomorphism of groups.
a) Show that if G is an infinite group and H is a finite group then the kernel ker(φ) := φ^-1({e}) = {g ∈ G | φ(g) = e} is nontrivial, i.e., ker(φ) ̸= {e}.
b) Show that if φ is an isomorphism of groups, then its inverse φ^-1 : H → G, φ(g) → φ^-1(φ(g)) = g, is also an isomorphism of groups.

Hi, can someone help me with this exercise, I really don't know how to prove it.

 

[Deleted member 4993]

Let φ : G → H be a homomorphism of groups.
a) Show that if G is an infinite group and H is a finite group then the kernel ker(φ) := φ^-1({e}) = {g ∈ G | φ(g) = e} is nontrivial, i.e., ker(φ) ̸= {e}.
b) Show that if φ is an isomorphism of groups, then its inverse φ^-1 : H → G, φ(g) → φ^-1(φ(g)) = g, is also an isomorphism of groups.

Hi, can someone help me with this exercise, I really don't know how to prove it.

Can you define homomorphism and kernel?

Please show us what you have tried and exactly where you are stuck.

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[blamocur]

A hint for a) : can you prove that there exist [imath]x,y\in G[/imath] such that [imath]x \neq y[/imath] and [imath]\phi(x) = \phi(y)[/imath] ?

Also, see https://www.freemathhelp.com/forum/threads/homomorphism-homework.135586/ for b).