Yes, mathematically. But it's really only true if the string has no mass at all.
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The period is independent of the mass.
The period of a pendulum is affected by the angle created by the swing of the pendulum, the length of the attachment to the mass, and the weight of the mass on the end of the pendulum.
As we derive the expression for period using dimensional analysis, we get T = 1/2pi ./(l/g) In this no mass is present. Hence the conclusion
In a simple pendulum, with its entire mass concentrated at the end of a string, the period depends on the distance of the mass from the pivot point. A physical pendulum's period is affected by the distance of the centre-of-gravity of the pendulum arm to the pivot point, its mass and its moment of inertia about the pivot point. In real life the pendulum period can also be affected by air resistance, temperature changes etc.
Yes. Given a constant for gravity, the period of the pendulum is a function of it's length to the center of mass. In a higher gravity, the period would be shorter for the same length of pendulum.