Gfrktyl

Suppose we have the circuit set up as shown in the image. Also suppose that $$V_s=overline{V_s}+v_S$$, where the smal v is for the variable component of the signal which varies much faster than the time constant of both capacitors. After some analisys we get that there's no gain in the variable component of the source voltage, that is, there's unitary gain, however from the equation of a bipolar transistor:
$$i_Eapprox i_c=beta i_s$$
thus there's a gain in current, how is it possible to hava a gain in current, and consequently a gain in power, but not in voltage? This seems to contradict Ohm's law, shouldn't a current gain be accompanied by a voltage gain?

Just because 2 quantities are related by an equation does not make them the same. For example, F=ma relatesd force and acceleration by a fixed parameter, mass, but they are still distinct concepts. Ohm's law is not a general rule, Kirchoff's laws are more general and do not imply that current and voltage are thje same: apply them to more complicated circuits than a single resistor and you would see this. Ohm's law has little to do with amplifiers. In the case of a voltage follwer, the output voltage is approximately the same as the input voltage but the available current is higher due to the current gain of the transistor. As already noted, the input resistance of the voltage follower is much higher than the output resistance so Ohm's law does not apply since these resistances are different.

... thus there's a gain in current, how is it possible to have a gain in current, and consequently a gain in power, but not in voltage?

By having a change in impedance.

In very crude terms the base current sees a resistance of βR_E whereas the output impedance is R_E.

This seems to contradict Ohm's law, shouldn't a current gain be accompanied by a voltage gain?

No. A transistor isn't a resistor.

This seems to contradict Ohm's law, shouldn't a current gain be accompanied by a voltage gain?

A transistor is an active device. It modulates the energy of an external power source to mimic the input signal. While Ohm's Law can be use to describe the behavior of certain aspects of active devices in some applications, keep in mind that it was derived to describe the relationships at play in a simple resistor. For anything beyond that you have to pay attention to the details.

By its definition, a bipolar transistor delivers current gain only. It is the circuit architecture and components around the transistor that translate this into voltage gain- sometimes.

Ic =hFE*Ib is then shared by both I(Re) and I(cout).

However for DC control, I(Cout) can never take all of the current of Ic . As Re serves a purpose to DC pull down the emitter and so R load must always be greater than Re for full swing voltages.

The result when overloaded, is the cap current pulls up the emitter so Vbe reduces then Ib reduces and the lower level signal cannot be reached.

Ohm’s Law still works here.
One simply uses Xc for the cap being much smaller so Rload must be greater than Re for linear operation

also Vce must be Vce saturation where hFE drops significantly as a another condition.

the power gain comes from hFE * Rb/Re equivalent loads