We call a property of a \(p\)-group \(G\) a group-theoretical invariant, if \(kG \cong kH\) implies that \(H\) has the same property. Here \(k\) denotes the field with \(p\) elements.
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.
‣ 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.
We document some of the major group-theoretical invariants for the MIP which are not easily available as standard GAP-functions.
‣ 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.
‣ 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 \(\sum_{g^G} \log_p(|C_G(g)/\Phi(C_G(g))| )\) where the sum runs over conjugacy classes of G. The next entries contain the number of conjugacy classe which are \(p^\ell\)-th powers, for \(0 \leq \ell \leq log_p(exp(G)) \) (Kuelshammer). Note that for \(\ell=0\) this is just the number of conjugacy classes in G. Finally, the following entries contain the number of conjugacy classes of \(p^\ell\)-th powers which are not central and have the same order as a class which powers to them where \(0 \leq \ell \leq log_p(exp(G))-1\) (Parmenter-Polcino Milies).
‣ 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.
‣ 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.
‣ MIPSandlingInfo ( G ) | ( operation ) |
For a given p-group G this function returns a list L containing information on the Sandling quotient \(G/\gamma_2(G)^p\gamma_3(G)\). The first entry describes \(Q = G/\gamma_2(G)^p\gamma_3(G)\) in the following way: If \(\gamma_2(G)^p\gamma_3(G) = 1\) and IdGroup is a known attribute of G, it is IdGroup(G). Otherwise it contains the GroupInfo of \(G/\gamma_2(G)^p\gamma_3(G)\) (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 \(G/\gamma_2(G)^p\gamma_4(G)\) in a similar way (Baginski/Margolis-Moede).
generated by GAPDoc2HTML