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Prove that if G is a finite group, the index of Z(G)

Chapter 7, Problem 38E

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QUESTION:

Prove that if G is a finite group, the index of Z(G) cannot be prime.

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QUESTION:

Prove that if G is a finite group, the index of Z(G) cannot be prime.

ANSWER:

Step 1 of 3

Suppose G be a finite group.

The aim of this problem is to prove that the index of Z(G) cannot be prime, that is, \([G: Z(G)]\) cannot be prime.

Step  2 of 3

Recall the G/Z Theorem: Suppose G be a group with center Z(G). If G/Z(G) is cyclic, then G

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