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# Groups

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Give an example of groups H_i, K_j such that H_1xH_2 is isomorphic to K_1xK_2 and no H_i is isomorphic to any K_j.

Let G be the additive group Q of rational numbers. Show that G is not the internal direct product of any two of its proper subgroups.

If G is the internal direct product of subgroups G_1 and G_2 show that G/G_1 is isomorphic to G_2 and G/G_2 is isomorphic to G_1.

I need detailed rigorous proof please. I have to study for a test.

https://brainmass.com/math/group-theory/groups-isomorphic-rational-numbers-230517

#### Solution Preview

Problem #1
We consider , , , , then we have
, but is not isomorphic to any .
Problem #2
Proof:
Suppose is the internal direct product of two subgroups and . Then we have and . Now we consider and it can be expressed as . Since both ...

#### Solution Summary

This provides examples of working with groups: isomorphic, direct products, and subgroups.

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