I am reading Undeniable Signatures by Chaum and Antwerpen. On page 2,
Consider using the group of known prime order $p$: All values transmitted between the participants are elements of this group, the multiplicatively denoted group operation is easily computed by all participants, ...
A suitable group of prime order $p$ and a primitive element $g$ are initially established and made public for use by a set of signers. Consider a particular signer S having a private key $x$ and a corresponding public key $g^{x}$. A message $m (\neq 1)$ is signed by S to form signature $z$, which should be equal to $m^{x}$. ... The initial challenge is of the form $z^{a}(g^{x})^{b}$, where verifier V chooses $a$ and $b$ independently and uniformly from the group elements.
Here, where is $x$ in? Is it an element in the group? What does $g^{x}$ mean? Is $m$ also an element in the group? What does $z^{a}$ mean? The exponent is not an integer so I am not sure what the expression mean. Thank you.