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Worksheet 11.3.1 Activity: Decomposing with Direct Products

Recall that last semester we saw that Z6Z2×Z3. When does this sort of thing happen?

1.

Given positive integers m and n, is it always true that ZmnZm×Zn? If this is not always true, for which m and n is it true? Try some (many) examples.

2.

Consider Z12. Can we break this down as the direct product of two smaller Zp groups? In other words is Z12=Zm×Zn for some values of m and n?

3.

Suppose your absent minded professor claims the answer is “no” and you don't feel like arguing. Maybe we can do something similar. Find two subgroups of Z12, call them H and K, such that HK={0} and HK=Z12. In general, HK={hk:hH,kK}; here it would be better to write H+K.


For any n, the group U(n) is the set of all positive integers less than and relatively prime to n, under multiplication modulo n. For example we saw that U(8)={1,3,5,7} is a group under multiplication modulo 8.

Consider the group U(28). The table below gives the twelve elements with their orders:

g 1 3 5 9 11 13 15 17 19 23 25 27
ord(g) 1 6 6 3 6 2 2 6 6 6 3 2

4.

Let G(n) be the set of all elements of order nk for some k (that is, elements with order some power of n). Find G(2) and G(3) for U(28).

5.

Are G(2) and G(3) subgroups of U(28)?

6.

Do G(2) and G(3) have the property that G(2)G(3)={1} and U(28)=G(2)G(3)?

7.

Is U(28)G(2)×G(3)? Is U(28)Zm×Zn for some values of m and n?