LagrangeEuler said:Homework Statement
Show that
##f(e)=e'## and ##f(g^-1)=f(g)^-1##
Homework Equations
Homomorphism
f(x\cdot y)=f(x)\cdot f(y)
The Attempt at a Solution
I show the first one. Neutral element is element which satisfied
##e\cdot e=e##.
So
##f(e)=f(e\cdot e=e)=f(e)\cdot f(e)##
So ##f(e)=e'##.
But how to show
##f(g^{-1})=f(g)^{-1}##?
A homomorphism of groups is a function between two groups that preserves the group structure. This means that the operation in the first group is still valid in the second group.
A homomorphism is a function that preserves the group structure, while an isomorphism is a bijective homomorphism. This means that an isomorphism not only preserves the group structure, but also preserves the elements and their relationships.
To determine if a function is a homomorphism of groups, you must check if it preserves the group operation. This means that for any two elements in the first group, their corresponding elements in the second group must also satisfy the group operation.
A homomorphism of groups can be either onto or one-to-one, or both. If the homomorphism is onto, it means that every element in the second group has at least one corresponding element in the first group. If the homomorphism is one-to-one, it means that each element in the second group has a unique corresponding element in the first group.
The kernel of a homomorphism of groups is the set of elements in the first group that are mapped to the identity element in the second group. In other words, it is the set of elements that are mapped to the neutral element under the homomorphism.