A Generalized Goursat Lemma

In this note the usual Goursat lemma, which describes subgroups of the direct product of two groups, is generalized to describing subgroups of a direct product $A_1\times A_2 \times…\times A_n$ of a finite number of groups. Other possible generalizations are discussed and applications characterizing several types of subgroups are given. Most of these applications are straightforward, while somewhat deeper applications occur in the case of profinite groups, cyclic groups, and the Sylow $p$-subgroups (including infinite groups that are virtual $p$-groups).

💡 Research Summary

The paper presents a comprehensive generalization of the classical Goursat Lemma, which characterizes subgroups of a direct product of two groups, to the case of a finite direct product of (n) groups (A_{1}\times A_{2}\times\cdots\times A_{n}). The authors begin by reviewing the historical development of Goursat’s result, noting its appearance in textbooks and various research articles, and they point out that a naïve extension by iterating the binary case runs into hidden complications when more than two factors are involved.

To overcome these difficulties, the authors introduce an asymmetric version of Goursat’s Lemma (Theorem 2.3). Instead of the usual symmetric quintuple ({G_{1},G_{1}’,G_{2},G_{2}’,\theta}), the asymmetric formulation uses a quadruple ({G_{1},G_{2},G_{2}’,\theta_{1}}) where (\theta_{1}:G_{1}\to G_{2}/G_{2}’) is a surjective homomorphism and the kernel of (\theta_{1}) coincides with the “bottom” subgroup (G_{1}’). The maps (Q_{2}) and (\Gamma_{2}) provide mutually inverse constructions between a subgroup of (A\times B) and its asymmetric data.

Armed with this tool, the authors develop an inductive construction for the (n)-fold product. They introduce the notation (G(j\mid S)) to denote the set of possible (j)-th coordinates of elements of a subgroup (G) when all coordinates indexed by the set (S) are forced to be the identity. Using this notation they define recursively a chain of intermediate subgroups (\Lambda_{i}) (the “partial projections”) and surjective homomorphisms (\theta_{i}) for (i=1,\dots,n-1). The main result (Theorem 3.2) states that there is a bijection between subgroups (G\le A_{1}\times\cdots\times A_{n}) and ((3n-2))-tuples

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