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Give an example to show that a direct product of two cyclic groups is not necessarily

Chapter 15, Problem 15.19

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QUESTION:

Give an example to show that a direct product of two cyclic groups is not necessarily cyclic. (Compare 15.18.)

Questions & Answers

QUESTION:

Give an example to show that a direct product of two cyclic groups is not necessarily cyclic. (Compare 15.18.)

ANSWER:

Step 1 of 4

In general, products of cyclic groups are not cyclic. If  is the cyclic group of order  is not cyclic.

The direct product of two cyclic groups,  is cyclic if and only if the

(where  is the order of one of the cycle groups, and  the order of the others.)

Note, we must have a Direct product of order .

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