Element Centralizers in the Centralizer Lattice

Part group theory, part universal algebra, we explore the centralizer operation on a group. We show that this is a closure operator on the power set of the group and compare it to the well-known closure operator of subgroup-generated-by'. We investigate properties of the centralizer lattice and consider the question of how generating sets should be addressed in this lattice. The element centralizers (centralizers of a single element) and, dually, their centers, play a fundamental role in the centralizer lattice. We show that every centralizer is a union of its centralizer equivalence classes’ over the element centers that it contains. We consider the Möbius function on the poset of element centers and obtain some new results regarding centralizers in a $p$-group.

💡 Research Summary

This paper presents a systematic study of the centralizer operation in group theory from a universal algebra perspective, focusing on its properties as a closure operator and the structure of the resulting lattice of centralizers.

The authors begin by establishing foundational properties of centralizers, demonstrating that the map (C_G(\cdot)) is order-reversing and interacts predictably with unions and intersections. A key initial observation is that (C_G(S) = C_G(\langle S \rangle)), confirming that the centralizer depends only on the subgroup generated by the set.

The core conceptual framework is developed in Section 3, where the pair ((C_G(\cdot), C_G(\cdot))) is shown to form a Galois connection on the power set (P(G)) of a group (G). Consequently, the composition (cl(\cdot) = C_G(C_G(\cdot))) is proven to be a closure operator, satisfying extensivity, monotonicity, and idempotence. This draws a direct analogy with the well-known closure operator “subgroup-generated-by” ((\langle \cdot \rangle)). The set of fixed points of this closure operator, denoted (\mathcal{C}(G)), is precisely the set of all centralizer subgroups in (G). The map (C_G(\cdot)) restricts to an order-reversing bijection on (\mathcal{C}(G)). The authors then define the centralizer lattice, where meet is given by intersection and join is defined as (H \vee K = C_G(C_G(H) \cap C_G(K))). They explore properties of the “fibers” of the centralizer map, showing that the union of all subsets (T) with (C_G(T) = C_G(S)) is exactly (C_G(C_G(S))).

The paper then shifts focus to the critical role played by element centralizers (C_G(g)). An equivalence relation (\sim) is defined on group elements by (x \sim y) if and only if (C_G(x) = C_G(y)). Using representatives (X) for these classes, the authors prove a fundamental decomposition theorem (Theorem 2): For any centralizer (H \in \mathcal{C}(G)) other than (Z(G)), there exist elements (g_1, …, g_t) such that the (C_G(g_i)) are all distinct proper element centralizers containing (H). Furthermore, (H) can be expressed as a disjoint union (H = (\bigcup_{i=1}^t Z^(g_i)) \cup Z(G)), where each (Z^(g_i)) is a subset of the center (Z(C_G(g_i))). This reveals the internal structure of centralizers as being assembled from building blocks related to element centers.

In the final section, the authors apply combinatorial tools to study centralizers in (p)-groups. They consider the poset formed by the centers of element centralizers, ({Z(C_G(g)) | g \in G}), under containment and define the Möbius function (\mu) on this poset. Their main result for (p)-groups (Theorem 3) states that for a nonabelian (p)-group and any centralizer (H \neq G), the sum of (\mu(Z(C_G(g)))) over all (g) satisfying (H \subseteq C_G(g) \subset G) is congruent to (-1) modulo (p). This connects the combinatorial invariants of the centralizer poset directly to the prime (p).

The paper also introduces the centralizer graph (\Gamma_Z(G)), whose vertices are the centralizers of noncentral elements, with an edge between (C_G(g)) and (C_G(h)) if (Z(C_G(h)) \subseteq C_G(g)). For the class of F-groups (where distinct element centralizers are incomparable under containment) that are also (p)-groups, they prove that every vertex in this graph has degree congruent to 0 modulo (p) (Theorem 4). The paper concludes by noting that this modular property does not hold for all (p)-groups, providing a distinction for this specialized class.

Overall, the work successfully bridges group-theoretic concepts with lattice theory and combinatorial enumeration, offering new structural insights into centralizers, particularly highlighting the foundational role of element centralizers and their centers within the broader lattice framework.