Let G be a finite group and p be a prime number such that p|o(G). Then any two sylow p-subgroup of G are conjugate .
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Let G be a finite group and p be a prime number such that p|o(G). Then any two sylow
p-subgroup of G are conjugate .
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- 43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.34. Suppose that and are subgroups of the group . Prove that is a subgroup of .9. Let be a group of all nonzero real numbers under multiplication. Find a subset of that is closed under multiplication but is not a subgroup of .