Generic classifications by p-group class - Revision history

Charles Eaton: Suzuki 2-groups

2024-01-09T11:15:50Z

Suzuki 2-groups

Charles Eaton: $C_{2^n} \wr C_2$

2023-10-23T17:00:50Z

Charles Eaton: /* Abelian 2-groups with 2-rank at most four */

2023-10-10T14:36:28Z

‎Abelian 2-groups with 2-rank at most four

Charles Eaton: /* Abelian 2-groups with 2-rank at most four */

2023-10-10T12:49:55Z

‎Abelian 2-groups with 2-rank at most four

Charles Eaton: /* Abelian 2-groups with 2-rank at most three */

2023-10-10T08:22:58Z

‎Abelian 2-groups with 2-rank at most three

Charles Eaton: Added extraspecial

2023-10-03T08:28:53Z

Added extraspecial

Charles Eaton: /* Fusion trivial p-groups */

2022-09-07T13:08:10Z

‎Fusion trivial p-groups

CesareGArdito: Added section about cyclic inertial quotient acting freely

2021-01-13T20:22:14Z

Added section about cyclic inertial quotient acting freely

Charles Eaton at 10:25, 15 February 2020

2020-02-15T10:25:26Z

Charles Eaton: Fusion trivial section created

2020-02-15T09:58:21Z

Fusion trivial section created

← Older revision		Revision as of 11:15, 9 January 2024
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	<math>PSU_3(q)</math> where <math>(q+1)_2=2^n</math>		<math>PSU_3(q)</math> where <math>(q+1)_2=2^n</math>
		+
		+	== Blocks whose defect groups are Suzuki <math>2</math>-groups ==
		+
		+	These blocks are described in [[References#E\|[Ea24]]]. Donovan's conjecture holds in this case. Suzuki <math>2</math>-groups are non-abelian <math>2</math>-groups <math>P</math> with more than one involution for which there is <math>\varphi \in {\rm Aut}(P)</math> permuting the involutions in <math>P</math> transitively. By [[References#\|[Hi63]]] these satisfy <math>\Omega_1(P)=Z(P)=\Phi(P)=[P,P]</math> and fall into classes A, B, C and D. The Sylow <math>2</math>-subgroups of the Suzuki simple groups and of <math>PSU_3(2^n)</math> fall in classes A and B respectively. Morita equivalence classes have representatives as follows:
		+
		+	a block of a finite group with a normal defect group
		+
		+	the principal block of <math>H</math> for <math>^2B_2(2^{2n+1}) \leq H \leq {\rm Aut}({}^2B_2(2^{2n+1}))</math> for some <math>n \geq 1</math>
		+
		+	a block of maximal defect of <math>H</math> where <math>Z(H) \leq [H,H]</math> and <math>PSU_3(2^n) \leq H/Z(H) \leq {\rm Aut}(PSU_3(2^n))</math> for some <math>n \geq 2</math> with <math>[H/Z(H):PSU_3(2^n)]</math> odd

← Older revision		Revision as of 17:00, 23 October 2023
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	These blocks are described in [[References#A\|[AE23]]]. Donovan's conjecture holds in this case.		These blocks are described in [[References#A\|[AE23]]]. Donovan's conjecture holds in this case.
		+
		+	== Principal blocks with wreath products <math>C_{2^n} \wr C_2</math> defect groups ==
		+
		+	These are classified up to splendid Morita equivalence in [[References#K\|[KoLaSa23]]]. There are six classes for each <math>n</math>, with representatives the principal blocks of:
		+
		+	<math>C_{2^n} \wr C_2</math>
		+
		+	<math>(C_{2^n} \times C_{2^n}):S_3</math>
		+
		+	<math>SL_2^n(q)</math> where <math>(q-1)_2=2^n</math>, consisting of matrices whose determinant is a <math>2^n</math> root of unity
		+
		+	<math>SU_2^n(q)</math> where <math>(q+1)_2=2^n</math>, consisting of matrices whose determinant is a <math>2^n</math> root of unity
		+
		+	<math>PSL_3(q)</math> where <math>(q-1)_2=2^n</math>
		+
		+	<math>PSU_3(q)</math> where <math>(q+1)_2=2^n</math>

@@ Line 72: / Line 72: @@
 |-
 |[[C(2^m)xC(2^m)xC2xC2|<math>C_{2^m} \times C_{2^m} \times C_2 \times C_2</math>]] ||  ||  || || [[References#E|[EL23]]] ||
 |}

@@ Line 32: / Line 32: @@
 == Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==
 These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.
-Let <math>l,m,n \geq 1</math> be distinct with  <math>l,m \neq 1</math>
+Let <math>l,m,n,r \geq 2</math> be distinct.
 {| role="presentation" class="wikitable mw-collapsible mw-collapsed"
-| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong>
+| <strong><math>rk_2(D) \leq 4</math> &nbsp;</strong>
 |-
 ! scope="col"| <math>D</math>
@@ Line 64: / Line 66: @@
 |-
 |[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]
 |}
 <!--== Abelian <math>2</math>-groups ==

@@ Line 31: / Line 31: @@
 Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].
-== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==
+== Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==
-These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.
+These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.
 Let <math>l,m,n \geq 1</math> be distinct with  <math>l,m \neq 1</math>

← Older revision		Revision as of 08:28, 3 October 2023
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	In each case, the Morita equivalence between the block and the class representative is known to be basic.		In each case, the Morita equivalence between the block and the class representative is known to be basic.
		+
		+	== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==
		+
		+	These blocks are described in [[References#A\|[AE23]]]. Donovan's conjecture holds in this case.

@@ Line 5: / Line 5: @@
 ''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.
-Blocks with fusion trivial defect groups must be nilpotent by [[References#P|[Pu88]]].
+Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].
 Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).
 Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.
 == Cyclic ''p''-groups ==

← Older revision		Revision as of 20:22, 13 January 2021
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	The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) \|principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives \|<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.		The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) \|principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives \|<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.
		+
		+
		+	== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group ==
		+
		+	<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E\|[ArMcK20]]].
		+
		+	If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>.
		+
		+	If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>.
		+
		+	In each case, the Morita equivalence between the block and the class representative is known to be basic.

@@ Line 1: / Line 1: @@
-This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.
+This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes.
 == Fusion trivial ''p''-groups ==
@@ Line 32: / Line 32: @@
 == Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==
 These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.
@@ Line 73: / Line 71: @@
 == Minimal nonabelian <math>2</math>-groups ==
 Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.
@@ Line 84: / Line 80: @@
 A Singer cycle is an element of order  <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.
-<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math>  contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial, or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n  \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>.
+<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math>  contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n  \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>.
 The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.

← Older revision		Revision as of 09:58, 15 February 2020
Line 1:		Line 1:
	This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.		This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.
		+
		+	== Fusion trivial ''p''-groups ==
		+
		+	''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V\|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.
		+
		+	Blocks with fusion trivial defect groups must be nilpotent by [[References#P\|[Pu88]]].
		+
		+	Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C\|[CG12]]] or [[References#S\|[Sa12b]]]).
		+
		+	Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S\|[St06]]] for an analysis of resistant ''p''-groups.

	== Cyclic ''p''-groups ==		== Cyclic ''p''-groups ==

Generic classifications by p-group class - Revision history

Charles Eaton: Suzuki 2-groups

Charles Eaton: C_{2^n} \wr C_2

Charles Eaton: /* Abelian 2-groups with 2-rank at most four */

Charles Eaton: /* Abelian 2-groups with 2-rank at most four */

Charles Eaton: /* Abelian 2-groups with 2-rank at most three */

Charles Eaton: Added extraspecial

Charles Eaton: /* Fusion trivial p-groups */

CesareGArdito: Added section about cyclic inertial quotient acting freely

Charles Eaton at 10:25, 15 February 2020

Charles Eaton: Fusion trivial section created

Charles Eaton: $C_{2^n} \wr C_2$