TITLE: Trivial group action
QUESTION [2 upvotes]: On page 43 of Dummit & Foote's abstract algebra:

Let $G$ be a group and $A$ a nonempty set.
Let $ga = a$, for all $g \in G$, $a \in A$. This action is called the trivial action and $G$ is said to act trivially on $A$. Note that distinct elements of $G$ induce the same permutation on $A$ (in this case the identity permutation). The associated permutation representation $G \to S_A$ is the
trivial homomorphism which maps every element of $G$ to the identity.

Question: Why does it say distinct elements of $G$? If $ga=a$, isn't $G$ the trivial group, which has only one element?

REPLY [2 votes]: $A$ is not a subgroup of $G$. It is neither a group (not necessarily)
So you can not say $ga=a\Rightarrow g=1_G$
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