Direct Product

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• Nov 24th 2009, 05:54 PM
bethh
Direct Product
Let G={1, 7, 17, 23, 49, 55, 65, 71} under multiplication modulo 96. Express G as an external and an internal direct product of cyclic groups.

I know that lGl =8, so G is isomorphic to either Z8, or G is isomorphic to the external direct product of Z4 Z2, or G is isomorphic to the external direct product of Z2 Z2 Z2. But I'm not sure what to do next.
• Nov 24th 2009, 07:09 PM
tonio
Quote:

Originally Posted by bethh
Let G={1, 7, 17, 23, 49, 55, 65, 71} under multiplication modulo 96. Express G as an external and an internal direct product of cyclic groups.

I know that lGl =8, so G is isomorphic to either Z8, or G is isomorphic to the external direct product of Z4 Z2, or G is isomorphic to the external direct product of Z2 Z2 Z2. But I'm not sure what to do next.

First, find out the orders of the different elements, for instance: $\displaystyle 7^2=49\,,\,7^3=343=55\!\!\!\!\pmod{96}\,,\,7^4=385 =1\!\!\!\!\pmod{96}\Longrightarrow\,ord(7)=4$ , and this already tells you that G can NOT be the direct product $\displaystyle \mathbb{Z}_2\times\mathbb{Z}\times\mathbb{Z}_2$ . Well, do the next ones now.

Tonio
• Nov 24th 2009, 08:10 PM
qmech
Patterns
In your problem, you can also notice a pattern:

49 = 48+1
55 = 48+7
65 = 48+17
71 = 48+23

so you might think this was { 1,7,17,23 } x { 0,1}.