[Paper Review] Group Structures on Families of Subsets of a Group

Mario Gómez, Sergio R. López-Permouth+3 more|arXiv (Cornell University)|Apr 5, 2016

Advanced Algebra and Logic3 citations

TL;DR

This paper introduces power groups—families of non-empty subsets of a group G that form a group under the subset product operation. It characterizes groups whose only power groups are subquotients (factor groups of subgroups), proving this occurs precisely when every element has finite order. The study reveals that not all power groups are isomorphic to subquotients, with ℚ serving as a counterexample where non-isomorphic power groups exist.

ABSTRACT

A binary operation on any set induces a binary operation on its subsets. We explore families of subsets of a group that become a group under the induced operation and refer to such families as power groups of the given group. Our results serve to characterize some types of groups in terms of their power groups. In particular, we consider when the only power groups of a group are the factor groups of its subgroups and when that is the case up to isomorphism. We prove that the former are precisely those groups for which every element has finite order and provide examples to illustrate that the latter is not always the case. In the process we consider several natural questions such as whether the identity element of the group must belong to the identity element of a power group or the inverse of an element in a power group must consist of the inverses of its elements.

Motivation & Objective

• To investigate families of subsets of a group G that form a group under the subset product operation.
• To characterize groups for which all power groups are isomorphic to subquotients (factor groups of subgroups).
• To explore the distinction between subquotients and a broader class of power groups called 'groups of cosets'.
• To determine conditions under which power groups are isomorphic to subquotients, and when they are not.
• To examine structural properties of power groups, such as whether the identity element must be a subgroup or contain the group identity.

Proposed method

• Define a power group as a family of non-empty subsets of a group G closed under the subset product operation ab = {gh | g∈a, h∈b}.
• Introduce two key classes: subquotients (factor groups of subgroups) and groups of cosets (induced by subsets E with E²=E and normality-like conditions).
• Use group-theoretic arguments to prove equivalence between four conditions: being a subquotient, having a subgroup as identity, closure under inverses in the power group, and forming a partition of a subgroup.
• Construct explicit examples, such as ℤ and ℚ, to demonstrate when all power groups are groups of cosets or when they are not.
• Apply the concept of 'underlying' groups: G₂ underlies G₁ if G₁ has a power group isomorphic to G₂.
• Use cardinality arguments to show that ℚ has a power group isomorphic to ℝ, hence not isomorphic to any subquotient.

Experimental results

Research questions

• RQ1 Which groups have the property that all their power groups are subquotients (i.e., factor groups of subgroups)?
• RQ2 Under what conditions is every power group of a group isomorphic to a subquotient?
• RQ3 Can a group have a power group that is not isomorphic to any subquotient?
• RQ4 Is the relation 'G₂ underlies G₁' transitive across groups?
• RQ5 What structural properties do groups share if one underlies another?

Key findings

• The only power groups of ℤ are groups of cosets, and all such power groups are isomorphic to subquotients.
• The additive group ℚ has a power group isomorphic to ℝ, which is uncountable and thus not isomorphic to any subquotient of ℚ.
• A power group is a subquotient if and only if its identity element is a subgroup of G.
• The identity element of a power group does not need to contain the identity of G, nor does it need to be a subgroup.
• Groups of cosets are not necessarily partitions, unlike subquotients, and can have identity elements that do not contain e.
• There exist non-isomorphic groups that underlie one another, such as ℚ and its power group isomorphic to ℝ.