Understanding semisimple Z modules

Question

I'm trying to understand the nature of semisimplicity in Z modules.

So for instance would I be right in thinking $Z/30Z \oplus Z/2Z$ is semi simple as it can be expressed as $Z/3Z \oplus Z/5Z \oplus Z/2Z \oplus Z/2Z$ ?

Similarly is $Z/20Z \oplus Z/7Z$ semi simple as it's expressed as $Z/2Z \oplus Z/2Z \oplus Z/5Z \oplus Z/7Z$

Are all prime order modules simple? So say $Z/101Z$ ?

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

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The only simple abelian groups are the cyclic groups of prime order (see here). Thus an abelian group is semisimple if and only if it is isomorphic to a sum of such groups. As a consequence, there is a unique semisimple abelian group of order $n$ for every integer $n$.

Also $Z/20Z$ is not isomorphic to $Z/2Z \oplus Z/2Z \oplus Z/5Z$ since the latter has no elements of order 4. In particular it is not semisimple.

edited Apr 13 '17 at 12:19

Community

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answered Feb 23 '16 at 21:37

Nitrogen

5,800

Thank you that clears things up a bit, but how do we know if a group is isomorphic to simple groups? – ForTheGlory Feb 23 '16 at 21:46
Just check if it is cyclic of prime order – Nitrogen Feb 23 '16 at 21:47
Sorry I don't understand what you mean, I mean how do we show a group is semisimple i.e isomorphic to product of simple groups – ForTheGlory Feb 23 '16 at 21:48
$A$ is semisimple iff for every prime $p$ dividing the order of $A$, there are no elements of order $p^k$ for $k>1$. Is that the kind of characterization you want? – Nitrogen Feb 23 '16 at 21:53
Yes! That's perfect thank you so much – ForTheGlory Feb 23 '16 at 21:57

Understanding semisimple Z modules

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