please solve 2(d) and explain what you do clearly. In this problem, we show that

Question

please solve 2(d) and explain what you do clearly.

In this problem, we show that any group of order 3 must be cyclic. Let G be a group of order 3, and let e, a, b be the three elements of G, where e is the identity of the group. Prove that b must be the inverse of a. [Hint: explain why ab must equal e.] Using part (a), prove that G must be cyclic. Consider the group G = {e, a, b, c, d. f, g, h} whose Cayley table is given below. What is the center of the group? What is |a| What is |b|? (Please show in some way how you found these.) Find a cyclic subgroup H of G such that |H | = 4. Find a noncyclic subgroup K of G such that |K| = 4.

Explanation / Answer

a) center of G={e,d} b)order of a=2 as a*a=a^2=e so |a|=2 now since b*b=d b*b*b=bd=g b*b*b*b=bg=e b^4=e |b|=4 c)cyclic subgroup={b,d,g,e}

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