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Does there exist an abelian $2$-group (an abelian group, all of whose elements have order a power $2$) of finite exponent that is not isomorphic to a direct sum of $2$-cyclic groups?

The exponent of G , denoted expG , is the smallest positive integer $m$ such that, for every $g\in G$ , $g^m=e_{G}$.

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    Please try to make your question self-contained, so that we know what you're asking without referring back to the title. And try to make it clearer too.2012-01-19
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    What exactly do you mean by "2-group"?2012-01-19
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    I am guessing that $2$-group means a group such that every element has order a power of $2$ (in recent questions Ali has asked about $p$-groups with this meaning). I am confused on "$2$-cyclic". Do you mean "cyclic $2$-groups"? Then you would be looking for counterexamples that are not finitely generated. (But they don't exist! See Arturo's answer.)2012-01-19
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    Please try to make the *posts* self-contained. Your post right now does not contain any questions, just a definition. The subject should not be an integral part of the post, without which it makes no sense, just like you shouldn't have to read the title of a book on the spine in order to understand the book.2012-01-19

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