Non-Abelian Tensor Products in Group Theory

The non-abelian tensor product of two groups captures the interaction between their commutator structures and mutual actions. Given groups G and H that act compatibly on each other, the non-abelian tensor product G ⊗ H is generated by symbols g ⊗ h subject to relations encoding both the group operations and the prescribed actions. This construction generalises the tensor product of abelian groups and connects directly to homotopical constructions such as the Whitehead product in algebraic topology. In algebraic terms, it provides a tool for analysing group extensions, evaluating Schur multipliers, and illuminating the lower central series. The special case of the tensor square G ⊗ G measures the degree of non-commutativity in G and has proved essential in the classification of finite p-groups, in computations within algebraic K-theory, and in investigations of finiteness and nilpotency bounds. Recent theoretical developments have extended the formalism to crossed modules and related higher-dimensional algebraic structures, yielding new explicit calculations and deepening our understanding of the interplay between group cohomology and tensorial constructions.

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Technical terms

Non-abelian tensor product: A group constructed from two groups with mutual actions, generated by symbols g ⊗ h subject to action-compatible relations.

Compatible actions: Mutual group actions on each other satisfying specific commutativity conditions required to define the non-abelian tensor product.

Tensor square: The special case G ⊗ G of a group with itself, measuring internal commutativity and linked to the lower central series.

Crossed module: An algebraic structure consisting of a group homomorphism with an action satisfying compatibility and Peiffer identities, used to generalise tensor products.

Schur multiplier: The second homology group H₂(G,ℤ), an invariant capturing extension and cohomological information that aids in computing tensor products.

References

1. A survey of non-abelian tensor products of groups and related constructions - doi: 10.5269/bspm.v30i1.13350. Boletim da Sociedade Paranaense de Matemática (2012).
2. Computing the nonabelian tensor squares of groups of order p3q. Arabian Journal of Mathematics (2021).
3. Compatible pair of actions for two same cyclic groups of 2-power order. Journal of Physics Conference Series (2017).
4. The non-abelian tensor product of normal crossed submodules of groups. Categories and General Algebraic Structures with Application (2020).