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The direct product of groups
and is denoted such that if then and also . -
For Abelian Groups, the direct sum is simply the direct product, but is denoted as
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(Fraleigh 11.2) The direct product of groups
and is a group. -
(Fraleigh e11.46) The direct sum of two Abelian groups is Abelian.
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Changing the order of the factors in a direct product yields a group isomorphic to the original one.
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(Fraleigh 11.9) If
and and are finite, then In general, if we have
and is finite, then -
is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Otherwise, it is indecomposable.