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Prove that a group of order 175 must have normal subgroups of orders 7 and 25. (See

Chapter 58, Problem 58.13

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QUESTION:

Prove that a group of order 175 must have normal subgroups of orders 7 and 25. (See Example 58.1.)

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QUESTION:

Prove that a group of order 175 must have normal subgroups of orders 7 and 25. (See Example 58.1.)

ANSWER:

Step 1 of 2

We have G to be a group of order 175. Now 175 can be factored as \(5^{2} \times 7\). Now, let us find out the number of Sylow 5-subgroups. The conditions to be satisfied are \(n_{5} \equiv 1(\bmod 5)\) and \(n_{5} \mid 7\). using these conditions we get \(n_{5}\) as 1.

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