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    After posting and receiving comments and answers, I realize this wasn't an appropriate question in the first place, for two reasons:

    1) The premise was false, as I overlooked the following subtle point: a central extension of abelian groups need not be abelian
    2) The answer was a basic fact in group cohomology (and I knew enough group cohomology to do algebraic number theory, but I hadn't, say, read an entire book on group cohomology)

    So, in a case like this, should I delete this question? I ask not only because of this particular question but because it has implications for MO in general.

    No, because there is an answer that (I assume) gives significant information. You can edit the question if you consider that appropriate, but IMO the edit should be as an addition to the original question. You can also vote to close and recommend in a comment that others vote to close if you consider that appropriate.

    • CommentAuthorAngelo
    • CommentTimeJan 5th 2013
    I agree with Bill. The question contained a false statement, but was still interesting, and elicited a good answer. I see no reason to delete it.