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If n 4 and is even, show that Dn has exactly n/2 noncyclic

Chapter 4, Problem 79E

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QUESTION:

If \(n \geq 4 \) and is even, show that \(D_n\) has exactly n/2 noncyclic subgroups of order 4.

Questions & Answers

QUESTION:

If \(n \geq 4 \) and is even, show that \(D_n\) has exactly n/2 noncyclic subgroups of order 4.

ANSWER:

Step 1 of 3

Let us consider thatis even.

The dihedral grouphasrotations namely andhasreflections namely.

It is given that is even, belongs to.

The subsets of  the form are closed under the composition.

Sois a non-cyclic subgroup of order four of, whenis even, and hereis a reflection.

 

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