On Posets of Classes of Automorphic Subgroups of Finite Groups

TL;DR

This paper introduces and analyzes the poset AutCl($G$) of automorphic classes of subgroups of finite groups, drawing parallels with the Iso$(G)$ framework. It proves that AutCl$(D_n)$ and AutCl$(Q_{4m})$ are distributive lattices, providing explicit structural descriptions and lattice-isomorphisms for several subfamilies, and shows how the automorphism groups influence the automorphic-class structure. It also completely characterizes finite groups for which AutCl$(G)$ is a chain, finding that only cyclic $p$-groups, elementary abelian $p$-groups, and $Q_8$ satisfy the condition. The work concludes with open problems and proposed generalizations, including extensions to autoprojectivities and conjectures on the poset structure for specific groups such as Heis$(\mathbb{Z}_p)$ for odd primes $p$.

Abstract

In [16], Tarnauceanu studied the poset Iso(G), of isomorphic classes of subgroups of a finite group G and proposed several questions for further research. In this paper, we study the poset AutCl(G), of classes of automorphic subgroups of finite group G. We introduce a partial order on AutCl(G) to tackle problem 5 mentioned in §4 of [16]. More precisely, we prove that AutCl(Dn) and AutCl(Q4m) are distributive lattices. Moreover, we characterize all classes of finite groups for which AutCl(G) is a chain.

Figures (9)

• Figure 1: $M_2$
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Lemma 2.1
• Remark 2
• proof
• Theorem 2.2
• proof
• Remark 3
• Theorem 2.3