Let g be a group and h a normal subgroup of finite index. suppose that p is a prime dividing [g: h]. prove that g has a subgroup such that h ⊆ k ⊆ g with [k: h] =p

24. a sports ball is inflated to an internal pressure of 1.85 atm at room temperature (25 °c). if the ball is then played with outside where the temperature is 7.5 °c, what will be the new pressure of the ball? assume the ball does not change in volume nor does any air leak from the ball a) 0.555 atm b) 1.74 atm c) 1.85 atm d) 1.97 atm

‘which reaction would most likely require the use of an inert electrode?

The ozone molecule o3 has a permanent dipole moment of 1.8×10−30 cm. although the molecule is very slightly bent-which is why it has a dipole moment-it can be modeled as a uniform rod of length 2.5×10−10 m with the dipole moment perpendicular to the axis of the rod. suppose an ozone molecule is in a 8000 n/c uniform electric field. in equilibrium, the dipole moment is aligned with the electric field. but if the molecule is rotated by a small angle and released, it will oscillate back and forth in simple harmonic motion.what is the frequency f of oscillation?