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2 Group theoretical invariants
 2.1 Computing Bins
 2.2 Some group-theoretical invariants

2 Group theoretical invariants

We call a property of a p-group G a group-theoretical invariant, if kG ≅ kH implies that H has the same property. Here k denotes the field with p elements.

2.1 Computing Bins

The following function applies the group theoretical invariants included in [MM20] to split the given groups into so-called bins. Groups that are in different bins do not share a certain group-theoretical invariant. In particular, they do not provide a counterexample to the MIP. The function also checks if a group lies in a class of groups for which the MIP is known based on the list in [MM20]. In this case it does not appear in any bin.

2.1-1 MIPBinsByGT
‣ MIPBinsByGT( p, n[, L] )( operation )

Given a list L of small group library ids or a list of groups of order p^n the function uses group theoretical invariants to split the groups into bins. If L is not given, then all groups of order p^n are considered.

2.2 Some group-theoretical invariants

We document some of the major group-theoretical invariants for the MIP which are not easily available as standard GAP-functions.

2.2-1 GroupInfo
‣ GroupInfo( G )( operation )

This is an auxiliary function used in other group-theoretical invariants. If IdGroup is available in GAP for the order of G it returns IdGroup(G). Otherwise it returns [Size(G), AbelianInvariants(G)]. This function remains unchanged from ModIsom, but was not documented before.

2.2-2 MIPConjugacyClassInfo
‣ MIPConjugacyClassInfo( G )( operation )

For a given p-group G this function returns a list L containing known group-theoretical invariants associated to the conjugacy classes of G. The first entry of L is the so-called Roggenkamp parameter ∑_g^G log_p(|C_G(g)/Φ(C_G(g))| ) where the sum runs over conjugacy classes of G. The next entries contain the number of conjugacy classe which are p^ℓ-th powers, for 0 ≤ ℓ ≤ log_p(exp(G)) (Kuelshammer). Note that for ℓ=0 this is just the number of conjugacy classes in G. Finally, the following entries contain the number of conjugacy classes of p^ℓ-th powers which are not central and have the same order as a class which powers to them where 0 ≤ ℓ ≤ log_p(exp(G))-1 (Parmenter-Polcino Milies).

2.2-3 SubgroupsInfo
‣ SubgroupsInfo( G )( operation )

For a given p-group G this function returns a list L containing at the i-th position the number of conjugacy classes of maximal elementary-abelian subgroups of order p^i in G (Quillen). This function remains unchanged from ModIsom, but was not documented before.

2.2-4 MIPJenningsInfo
‣ MIPJenningsInfo( G )( operation )

For a given p-group G this function returns a list L containing information on quotients of the Jennings-Zassenhaus series D_i(G) of G. Starting with i=1 for increasing i it contatins the GroupInfo for D_i(G)/D_i+1(G), D_i(G)/D_i+2(G), and D_i(G)/D_2i+1(G) when these are defined. The last entry describes G/D_3(G) if p=2 and G/D_4(G) if p>2. If D_3(G)=1 or D_4(G)=1, respectively, and IdGroup is a known attribute of G, it is IdGroup(G). Otherwise it contains the GroupInfo of G/D_3(G)=1 or G/D_4(G)=1 respectively.

2.2-5 MIPSandlingInfo
‣ MIPSandlingInfo( G )( operation )

For a given p-group G this function returns a list L containing information on the Sandling quotient G/γ_2(G)^pγ_3(G). The first entry describes Q = G/γ_2(G)^pγ_3(G) in the following way: If γ_2(G)^pγ_3(G) = 1 and IdGroup is a known attribute of G, it is IdGroup(G). Otherwise it contains the GroupInfo of G/γ_2(G)^pγ_3(G) (Sandling). Moreover, if G is generated by at most two elements and the length of the Jennings-Zasenhaus series of G is at least four, it contains a second entry describing G/γ_2(G)^pγ_4(G) in a similar way (Baginski/Margolis-Moede).

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