Difference between revisions of "MNA(2,1)"
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== Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> == | == Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> == | ||
| − | The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but there exists | + | The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>. By [[References#S|[Sa16]]] all non-nilpotent blocks with this defect group are [[Glossary#Isotypy|isotypic]]. |
| + | '''<pre style="color: red">CLASSIFICATION NOT COMPLETE</pre>''' | ||
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! scope="col"| Class | ! scope="col"| Class | ||
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| − | |[[M(16, | + | |[[M(16,3,1)]] || <math>k(MNA(2,1))</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 || |
|- | |- | ||
| − | |[[M(16, | + | |[[M(16,3,2)]] || <math>B_0(k(A_5:C_4))</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 || |
| − | |} | + | |- |
| + | |[[M(16,3,3)]] || <math>k(A_4:C_4)</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 || | ||
| + | |} | ||
| + | |||
| + | If <math>B</math> is not nilpotent, then <math>k(B)=10, k_1(B)=2, l(B)=2</math>. | ||
Latest revision as of 11:47, 9 November 2022
Blocks with defect group [math]MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math]
The defect groups are minimal nonabelian [math]2[/math]-groups. The invariants [math]k(B)[/math], [math]l(B)[/math] and [math]k_i(B)[/math] for all [math]i[/math] are determined in [Sa11]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the CFSG. The automorphism group of [math]MNA(2,1)[/math] is a [math]2[/math]-group, but by [Sa14,12.7] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) [math]\cong A_4:C_4[/math]. By [Sa16] all non-nilpotent blocks with this defect group are isotypic.
CLASSIFICATION NOT COMPLETE
| Class | Representative | # lifts / [math]\mathcal{O}[/math] | [math]k(B)[/math] | [math]l(B)[/math] | Inertial quotients | [math]{\rm Pic}_\mathcal{O}(B)[/math] | [math]{\rm Pic}_k(B)[/math] | [math]{\rm mf_\mathcal{O}(B)}[/math] | [math]{\rm mf_k(B)}[/math] | Notes |
|---|---|---|---|---|---|---|---|---|---|---|
| M(16,3,1) | [math]k(MNA(2,1))[/math] | 1 | 10 | 1 | [math]1[/math] | 1 | 1 | |||
| M(16,3,2) | [math]B_0(k(A_5:C_4))[/math] | ? | 10 | 2 | [math]1[/math] | 1 | 1 | |||
| M(16,3,3) | [math]k(A_4:C_4)[/math] | ? | 10 | 2 | [math]1[/math] | 1 | 1 |
If [math]B[/math] is not nilpotent, then [math]k(B)=10, k_1(B)=2, l(B)=2[/math].
