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Recently Aragona et al. have introduced a chain of normalizers in a Sylow2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup.They have shown that the indices of consecutive groups in the chain depend onthe number of partitions into distinct parts and have given a description, bymeans of rigid commutators, of the first n-2 terms in the chain. Moreover, theyproved that the (n-1)-th term of the chain is described by means of rigidcommutators corresponding to unrefinable partitions into distinct parts.Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n),for p > 2 computing the chain of normalizers becomes a challenging task, in theabsence of a suitable notion of rigid commutators. This problem is addressedhere from an alternative point of view. We propose a more general framework forthe normalizer chain, defining a chain of idealizers in a Lie ring over Z_mwhose elements are represented by integer partitions. We show how thecorresponding idealizers are generated by subsets of partitions into at mostm-1 parts and we conjecture that the idealizer chain grows as the normalizerchain in the symmetric group. As an evidence of this, we establish acorrespondence between the two constructions in the case m=2.

A modular idealizer chain and unrefinability of partitions with repeated parts