Examples of Non-Faithful Group Actions

Question

I cannot find anywhere a relatively simple example of a non-faithful group action.

I feel I understand the definition relatively well, however I can't come up with any ideas for one in my head (and despite scouring the internet, the only ones I have seen are for groups which, in my introductory group theory class, we have not covered).

Are there any simple examples that anyone can suggest?

Answer 1

Take any group $H$ acting faithfully on a set $X$ and any noninjective group homomorphism $G\to H$. Then $G$ acts on $X$ as well, but not faithfully.

This may sound contrived, but actually any non-faithful action is of this kind (we can simply let $H$ be $G$ modulo the kernel of the action).

Answer 2

For a concrete example, groups with a non-trivial center acting on their set of subgroups via conjugation work well (in fact, I think Hagen von Eitzen gave you a description of all non-faithful group actions).

In the dihedral groups acting on a regular $2n$-gon with $n$ even, the half-rotation is in the center of the group, and fixes every subgroup.

Answer 3

The group $SL_2(\mathbb{R})$ has a natural but nonfaithful action on the projective space $\mathbb{R}P^1$. Letting $M \in SL_2(\mathbb{R})$ and letting $\ell_{\vec v} \in \mathbb{R}P^1$ be the line through the origin with a direction vector $\vec v$, the action is given by $$M \cdot \ell_{\vec v} = \ell_{M \vec v} $$ The kernel of this action is all multiples of the identity matrix $\pmatrix{a&0\\0&a}$.