Let G = [a] be a cyclic group of order 16. Let H = [a^4]. a. List the elements o

Question

Let G = [a] be a cyclic group of order 16.  Let H = [a^4].

a.   List the elements of H, then give the index of H in G.

b.   Explain why H must be a normal subgroup in G without explicily checking cosets.

c.   List the elements of the quotient (factor) group G/H and construct the quotient group's operation table.

d.   Determine if G/H is a cyclic group.  If not cyclic, explain why.  If cyclic give a generator of G/H.

e.   Let ?: G onto G/H be the natural homomorphism map from G to G/H.  Illustrate the application of the natural map on the elements a^2 and a^5 from G.  Then list the kernal of this map.

f.   Is the natural map ?: G onto G/H an isomorphism? Explain

Let G = [a] be a cyclic group of order 16. Let H = [a^4]. List the elements of H, then give the index of H in G. Explain why H must be a normal subgroup in G without explicily checking cosets. List the elements of the quotient (factor) group G/H and construct the quotient group's operation table. Determine if G/H is a cyclic group. If not cyclic, explain why. If cyclic give a generator of G/H. Let ?: G onto G/H be the natural homomorphism map from G to G/H. Illustrate the application of the natural map on the elements a^2 and a^5 from G. Then list the kernal of this map. Is the natural map ?: G onto G/H an isomorphism? Explain

Explanation / Answer

Let G = [a] be a cyclic group of order 16. Let H = [a^4]. a. List the elements o

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