Let G = [a] be a cyclic group of order 16. Let H = [a^4]. a. List the elements o
ID: 2970368 • Letter: L
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? ExplainExplanation / Answer
Let G = [a] be a cyclic group of order 16. Let H = [a^4]. a. List the elements o
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