Question

12pts) 1. Both Lagrange's theorem and Cauchy's theorem deal with the relationship between the size of a group and the order of its elements. (a) Explain the difference between the theorems in general terms and by using S, as an example. Your explanation should include what we can and cannot conclude from each theorem about S7. (b) Which theorem would allow you to prove that if a group contained only elements that had order some power of 2, then the order of the group must be a power of 2? You might want to consider contrapositives. Then give the proof. (c) We have previously proved that if G is a cyelic group of order n, then for every factor m of n, G contains elements of order m. Explain why neither Lagrange's theorem nor Cauchy's theorem can be used to prove this.

Answer 1

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