Paper 3, Section II, D

Algebra and Geometry | Part IA, 2007

Let xx be an element of a finite group GG. What is meant by the order of xx ? Prove that the order of xx must divide the order of GG. [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]

If GG is a group of order nn, and dd is a divisor of nn with d<nd<n, is it always true that GG must contain an element of order dd ? Justify your answer.

Prove that if mm and nn are coprime then the group Cm×CnC_{m} \times C_{n} is cyclic.

If mm and nn are not coprime, can it happen that Cm×CnC_{m} \times C_{n} is cyclic?

[Here CnC_{n} denotes the cyclic group of order nn.]

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