Does a Trivial Group Action Mean Every Element Maps to Itself?

[JFo]

When it's said that a group G acts "trivially" on a set X does that mean

$$\forall g \in G, \forall x \in X, gx = x$$

??

 

[matt grime]

Yes, that is what it means.

 

[tacman]

Yes, when it's said that a group G acts "trivially" on a set X, it means that for every element g in G and every element x in X, the action of g on x results in x itself. This means that the group does not have any non-trivial effects on the elements of the set, and the action is essentially equivalent to the identity element of the group. In other words, the group is not changing or transforming the elements of the set in any way. This is often referred to as the trivial action because it is a very simple and uninteresting action compared to other actions that a group may have on a set.