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nLab > Latest Changes: perfect group

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeOct 27th 2018
- PermaLink
Author: Urs
Format: MarkdownItexpolished layout, added example of $2I$ <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/2">v2</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

polished layout, added example of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">2I</annotation></semantics></math>$

diff, v2, current
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeOct 27th 2018
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexAdded example of $SL_n(F)$, and that quotients of perfect groups are perfect. <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/3">v3</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

Added example of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_n(F)</annotation></semantics></math>$ , and that quotients of perfect groups are perfect.

diff, v3, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeOct 27th 2018
- (edited Oct 27th 2018)
- PermaLink
Author: Urs
Format: MarkdownItexmade 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? <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/4">v4</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

made explicit that the class of examples $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>𝔽</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_n(\mathbb{F})</annotation></semantics></math>$ generalizes the example $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>I</mi><mo>≃</mo><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>5</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2I \simeq SL_2(\mathbb{F}_5)</annotation></semantics></math>$

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

diff, v4, current
- CommentRowNumber4.
- CommentAuthorTodd_Trimble
- CommentTimeOct 27th 2018
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexAdded two references for the $SL_n$ proposition, and another proposition that perfect groups are closed under colimits in $Grp$. <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/5">v5</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

Added two references for the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">SL_n</annotation></semantics></math>$ proposition, and another proposition that perfect groups are closed under colimits in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math>$ .

diff, v5, current
- CommentRowNumber5.
- CommentAuthorTodd_Trimble
- CommentTimeOct 28th 2018
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexHm, 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 $60$.

Hm, something that seems kind of cool is the coincidence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>5</mn></msub><mo>≅</mo><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mrow><msup><mn>2</mn> <mn>2</mn></msup></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_5 \cong SL_2(\mathbb{F}_{2^2})</annotation></semantics></math>$ (the latter group is of order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msup><mn>4</mn> <mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mn>4</mn> <mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>4</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">(4^2 - 1)(4^2 - 4)/(4-1) = 60</annotation></semantics></math>$ ). I haven’t looked at this carefully, but it must follow from the statement that there is a unique perfect group of order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>60</mn></mrow><annotation encoding="application/x-tex">60</annotation></semantics></math>$ .
- CommentRowNumber6.
- CommentAuthorTodd_Trimble
- CommentTimeOct 28th 2018
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexPut 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 $\mathbb{F}_4$!! <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/6">v6</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

Put in the correct theorem numbers from Lang’s Algebra. He proves that (except for the case $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = \mathbb{Z}/(2)</annotation></semantics></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(3)</annotation></semantics></math>$ ) the group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_n(F)</annotation></semantics></math>$ is not only perfect but also that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>PSL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_n(F)</annotation></semantics></math>$ is simple. For the case $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>4</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>4</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>±</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{F}_4) = SL_2(\mathbb{F}_4)/\pm I</annotation></semantics></math>$ , this does not mean there’s a simple group of order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>30</mn></mrow><annotation encoding="application/x-tex">30</annotation></semantics></math>$ , because in fact $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>=</mo><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 = -1</annotation></semantics></math>$ in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>𝔽</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_4</annotation></semantics></math>$ !!

diff, v6, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeOct 28th 2018
- PermaLink
Author: Urs
Format: MarkdownItexThanks! 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]]_.

Thanks! I have made more formal pointer to the refernces. And copied over the statement of perfection of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">SL_n</annotation></semantics></math>$ over fields to special linear group.
- CommentRowNumber8.
- CommentAuthorTodd_Trimble
- CommentTimeOct 28th 2018
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexGave 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. <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/9">v9</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

Gave a simple argument for why $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>4</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_2(\mathbb{F}_4)</annotation></semantics></math>$ is “the” simple group of order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>60</mn></mrow><annotation encoding="application/x-tex">60</annotation></semantics></math>$ (“the” in parentheses since there are nontrivial automorphisms); this and not $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>5</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_2(\mathbb{F}_5)</annotation></semantics></math>$ is the smallest example from the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SL</mi></mrow><annotation encoding="application/x-tex">SL</annotation></semantics></math>$ 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
- PermaLink
Author: Urs
Format: MarkdownItexadded the example ([here](https://ncatlab.org/nlab/show/perfect+group#TerminalEpimorphismsInInfinityGroupoids)) of $\mathbf{B}G \to \ast$ being epi in $Grpd_\infty$ if $G$ is perfect. <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/10">v10</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

added the example (here) of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math>$ being epi in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">Grpd_\infty</annotation></semantics></math>$ if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ is perfect.

diff, v10, current
- CommentRowNumber10.
- CommentAuthorTim_Porter
- CommentTimeDec 16th 2021
- PermaLink
Author: Tim_Porter
Format: MarkdownItexlinked to [[Quillen plus construction ]] <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/12">v12</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

linked to Quillen plus construction

diff, v12, current
- CommentRowNumber11.
- CommentAuthorTim_Porter
- CommentTimeDec 16th 2021
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Author: Tim_Porter
Format: MarkdownItexand [[Steinberg group]]. <a href="https://ncatlab.org/nlab/revision/diff/perfect+group/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/perfect+group/12">v12</a>, <a href="https://ncatlab.org/nlab/show/perfect+group">current</a>

and Steinberg group.

diff, v12, current

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nLab > Latest Changes: perfect group