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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 27th 2018

polished layout, added example of $2I$

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeOct 27th 2018

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

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

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

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

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 27th 2018

Added two references for the $SL_n$ proposition, and another proposition that perfect groups are closed under colimits in $Grp$.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 27th 2018

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

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeOct 27th 2018

Put in the correct theorem numbers from Lang’s Algebra. He proves that (except for the case $n=2$ and $F = \mathbb{Z}/(2)$ or $\mathbb{Z}/(3)$) the group $SL_n(F)$ is not only perfect but also that $PSL_n(F)$ is simple. For the case $PSL_2(\mathbb{F}_4) = SL_2(\mathbb{F}_4)/\pm I$, this does not mean there’s a simple group of order $30$, because in fact $1 = -1$ in $\mathbb{F}_4$!!

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 28th 2018

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

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeOct 28th 2018

Gave a simple argument for why $SL_2(\mathbb{F}_4)$ is “the” simple group of order $60$ (“the” in parentheses since there are nontrivial automorphisms); this and not $SL_2(\mathbb{F}_5)$ is the smallest example from the $SL$ 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.