Let a and b be elements of a finite group G. Prove that a and a -1 have the same

Question

Let a and b be elements of a finite group G. Prove that a and a -1 have the same order. Prove that a and bab -1 have the same order. Prove that ab and ba have the same order. Let G be a group and define the relation R on G by if and only if a and b the same order. Prove that R is an equivalence relation. Prove that a subset H of a finite group G is a subgroup of G if and only if H is nonempty, and a H and b H imply ab H, (Hint: Use Corollary 3.16) In Exercise 17 of Section 3.3, the center Z(G) is defined as Z(G) = {a G | ax = xa for every x G}. Prove that if b is the only element of order 2 in G, then b Z(G). If a is an element of order m in a group G and ak = e, prove that m divides k. If G is a cyclic group, prove that the equation x2 = e has not most two distinct solution in G. Let G be a finite cyclic group of order n. If d is a positive divisor of n, prove that

Explanation / Answer

I assume you've seen that the map g (x) = gxg-1 is an automorphism. Also, automorphisms take elements of G to an element of the same order. Because b is the only element of order 2, g(b) = b for all g in G. This means that for all g in G, gbg-1 = b, or that gb = bg. Hope this helps.

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