On constructing CAP-subgroups in direct products
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Archiv der Mathematik
On constructing CAP-subgroups in direct products Joseph Petrillo
Abstract. The cover-avoidance property and its variations have become central in the study of finite groups. Along with this widespread attention comes a need to construct examples and counterexamples. The main goal of this work is to apply the theory of direct products to systematically facilitate such constructions. Mathematics Subject Classification. Primary 20E07; Secondary 20E15. Keywords. Direct product, Cover-avoidance property, CAP-subgroup.
1. Introduction. All groups considered in this paper are finite. The cover-avoidance property has earned a central role in the study of finite groups. What began as a tool for understanding the deeper structure of solvable groups [8,15,17] has become a main area of study in classifying groups of various types [7,9,11]. In recent years, both stronger and weaker versions of covering and avoidance have also been studied extensively by numerous authors, including [1–3]. Along with this widespread attention comes a need to construct examples of subgroups satisfying cover-avoidance conditions. The main goal of this work is to systematically investigate such constructions using direct products, which also hold widespread interest (for instance, see [4–6]). In [12], the author presents necessary and sufficient conditions for a subgroup in a direct product of finite groups to possess the cover-avoidance property. However, the present focus is on refining those results for the purpose of practical applications. In addition, the results presented here will be of fundamental interest to those studying direct products of finite groups since our work will require some deeper insight into the underlying normal subgroup structure. 2. Preliminary results. The subgroup structure of a direct product G1 × G2 is well-known and credited to Goursat (see [16, Theorem 1.6.1]).
J. Petrillo
Arch. Math.
Theorem 2.1. The subgroups of a direct product G1 × G2 are in one-to-one correspondence with triples (S1 , S2 , φ) where S1 is a section of G1 , S2 is a section of G2 , and φ : S1 → S2 is an isomorphism. In particular, if U is a subgroup of G1 × G2 corresponding to (S1 , S2 , φ), then U ∩ Gi πi (U ) and Si = πi (U )/(U ∩ Gi ), where πi denotes the natural projection of G1 × G2 onto Gi , i ∈ {1, 2}. If U is a subgroup of G1 × G2 corresponding to (S1 , S2 , φ), then U can be identified with the set of all pairs (u1 , u2 ) in π1 (U ) × π2 (U ) such that φ(u1 (U ∩ G1 )) = u2 (U ∩ G2 ). We will not distinguish between the internal and external viewpoints of a direct product. For instance, the subgroups πi (U ) and U Gj ∩ Gi , i = j, will be considered identical. Following [11] and [12], if the subgroup U of G1 × G2 is such that U = U1 × U2 for some U1 ≤ G1 and U2 ≤ G2 , then we call U a non-diagonal subgroup of G1 × G2 . In this case, the sections S1 and S2 of U are both trivial and hence πi (U ) = U ∩ Gi for i ∈ {1, 2}. Otherwise we shall call U a diagonal subgroup. Now consider a chief factor H/K of G1 × G2 , and n
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