frigidaire glasstop stove element turns off and on during use

Seems normal burner will heat up and cool down to maintain temp

It is possible that the element that you purchased has a defect in it. The element may act as if it has continuity for electrical current to flow through it when cold. As it heats up there is some expansion going on and this could cause some part of the resistive electrical conductor to open circuit and interrupt the current flow till the unit cools back down some.

Since it is possible that this problem could also be in the electrical attachment to the heater element I would suggest that you carefully check the connections to make sure things are tight and clean with no corrosion. If nothing is found and the problem persists then return the element for a replacement.

Smooth-top (ceramic) stoves can be damaged if an element gets too hot. To prevent this, each element has a thermostat which cuts off the heat at a certain temperature. It probably won't happen if you have a properly-sized pot of water heating up, but if you put the element on high with no pot then the element will definitely start cycling off and on at some point.