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nLab > Latest Changes: perfect group

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2018

    polished layout, added example of 2I2I

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 27th 2018

    Added example of SL n(F)SL_n(F), and that quotients of perfect groups are perfect.

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2018
    • (edited Oct 27th 2018)

    made explicit that the class of examples SL n(𝔽)SL_n(\mathbb{F}) generalizes the example 2I≃SL 2(𝔽 5)2I \simeq SL_2(\mathbb{F}_5)

    What’s a good reference to cite for all these facts and examples?

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 27th 2018

    Added two references for the SL nSL_n proposition, and another proposition that perfect groups are closed under colimits in GrpGrp.

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 28th 2018

    Hm, something that seems kind of cool is the coincidence A 5≅SL 2(𝔽 2 2)A_5 \cong SL_2(\mathbb{F}_{2^2}) (the latter group is of order (4 2−1)(4 2−4)/(4−1)=60(4^2 - 1)(4^2 - 4)/(4-1) = 60). I haven’t looked at this carefully, but it must follow from the statement that there is a unique perfect group of order 6060.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 28th 2018

    Put in the correct theorem numbers from Lang’s Algebra. He proves that (except for the case n=2n=2 and F=ℤ/(2)F = \mathbb{Z}/(2) or ℤ/(3)\mathbb{Z}/(3)) the group SL n(F)SL_n(F) is not only perfect but also that PSL n(F)PSL_n(F) is simple. For the case PSL 2(𝔽 4)=SL 2(𝔽 4)/±IPSL_2(\mathbb{F}_4) = SL_2(\mathbb{F}_4)/\pm I, this does not mean there’s a simple group of order 3030, because in fact 1=−11 = -1 in 𝔽 4\mathbb{F}_4!!

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2018

    Thanks! I have made more formal pointer to the refernces. And copied over the statement of perfection of SL nSL_n over fields to special linear group.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 28th 2018

    Gave a simple argument for why SL 2(𝔽 4)SL_2(\mathbb{F}_4) is “the” simple group of order 6060 (“the” in parentheses since there are nontrivial automorphisms); this and not SL 2(𝔽 5)SL_2(\mathbb{F}_5) is the smallest example from the SLSL class.

    Another thing one can say is that finite cartesian products of perfect groups and infinite direct sums of perfect groups are perfect. (In both cases the argument is very simple.) But I don’t know what one can say for infinite cartesian products – I wasn’t able to google my way to an answer.

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2021

    added the example (here) of BG→*\mathbf{B}G \to \ast being epi in Grpd ∞Grpd_\infty if GG is perfect.

    diff, v10, current

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeDec 16th 2021
    • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeDec 16th 2021

1 to 11 of 11

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