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Ring with Z as its group of units?

Given a group $G$, does there always exist a ring $R$ such that $R^\times \cong G$? I feel like this isn't true but that's just a hunch. One of my friends asked me this yesterday and we couldn't come up with an answer so I thought I'd ask here.

  • $\begingroup$ Does the group ring $F_2[G]$ work? $\endgroup$ – George Lowther Oct 19 '11 at 0:18
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    $\begingroup$ This came up before. See mathoverflow.net/questions/75192/… $\endgroup$ – Faisal Oct 19 '11 at 0:24
  • $\begingroup$ And Jesse Elliot's comment says that $F_2[G]$ does not work. $\endgroup$ – George Lowther Oct 19 '11 at 0:27
  • $\begingroup$ When $|G|$ is finite and odd, then it must be the direct product of cyclic groups of order $2^k-1$ for various $k$; in other words, you only have to worry about the groups coming from $R=GF(2^{n_1})\times GF(2^{n_2})\times\cdots$. $\endgroup$ – Steve D Oct 19 '11 at 0:45
  • $\begingroup$ Nice question! $\endgroup$ – Anton Geraschenko Oct 19 '11 at 1:21