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3 Jennings bound
 3.1 Jennings bound

3 Jennings bound

3.1 Jennings bound

For a pair of groups G and H the Jennings bound is defined as the maximal integer s such that G/D_s(G) ≅ H/D_s(H), where D_i is the Jennings-Zassenhaus series (see [MM20]). If s is the Jennings bound for G and H, then it follows that I(kG)/I(kG)^s ≅ I(kH)/I(kH)^s. Thus s is a minimum for the level until which MIPBinSplit needs to run to be able to split the groups.

3.1-1 JenningsBound
‣ JenningsBound( p, n, L )( operation )

Given a list L of small group library ids or a list of groups of order p^n the function computes an integer b such that the quotients of the associated augmentation ideals are guaranteed to be isomorphic up to class b-1.

More precisely b is the biggest integer such that for any G, H ∈ L one has G/D_b(G) ≅ H/D_b(H).

3.1-2 JenningsBoundPairwise
‣ JenningsBoundPairwise( p, n, L )( operation )

Given a list L of small group library ids or a list of groups of order p^n the function computes for all pairs (G,H) of groups in the list an integer b such that the quotients of the associated augmentation ideals are guaranteed to be isomorphic up to class b-1.

More precisely the return is a list of triples. The first two entries of each triple are two groups G and H, or their ids if they are available, and the last entry contains JenningsBound(p,n,[G,H]).

3.1-3 JenningsBoundConjecture
‣ JenningsBoundConjecture( p, n, L )( operation )

Given a list L in the format returned by MIPBinSplit for some groups of order p^n this function checks if the groups violate the bound conjectured in Question 2.7 [MM20] on the maximal quotients of the corresponding augmentation ideals which need to be checked to decide MIP.

For elements of MIPResults(p,n) which are solved by theoretical results, or which remain open, it returns fail.

3.1-4 JenningsBoundConjectureIsStrict
‣ JenningsBoundConjectureIsStrict( p, n, L )( operation )

Given a list L in the format returned by MIPBinSplit for some groups of order p^n this function checks if the groups attain the bound conjectured in Question 2.7 [MM20] on the maximal quotients of the corresponding augmentation ideals which need to be checked to decide MIP. This function currently only works if L contains the information for a pair of groups.

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