- #1

jgoldstein

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## Homework Statement

show for any group G, the function G x G --> G defined by (h)^g = hg is a faithful right action of the group G on itself. In this case, G is said to act on itself by right multiplication.

## Homework Equations

Definition of faithful: a right action of a group g on a set X is said to be faithful if the only element of G that acts trivially on the entire set is the identity element. So an action is faithful if x^g = x for all x in X i.e g = e, the identity element

## The Attempt at a Solution

My professor did this in class and I am a bit confused.

Suppose (h)^g1 = (h)^g2 for all h in G. Then by definition hg1 = hg2.

Then we left multiply by h^(-1) and we have g1=g2

I do not understand how this shows faithful right action?

Can someone explain?