Are the following definitions of conjugates related?

I have seen the following two definitions of conjugates and I'm wondering if these definitions are related somehow?

In "A Book of Abstract Algebra" by Pinter, the definition is

Let $G$ be a group (with operation $*$) and $a, x \in G$. then $x * a * x^{-1}$ is called a conjugate.

In "Linear Algebra Done Right" by Axler, the definition is

For the real field, the conjugate of $x \in \mathbb{R}$ is $x$. For the complex field, the conjugate of $(a, b) \in \mathbb{C}$ is $(a, -b)$.

Solution 1:

In both cases is is the image under a group action.

In the first case the action of $G$ on itself, the image of $a$ when acting by $x$.

In the second case is the action of the Galois automorphism "complex conjugation" on the complex numbers.