Can a group be isomorphic to a proper subgroup of itself? For finite groups, the answer is clearly no- such a structure would contradict cardinality constraints. But in the realm of infinite groups, the question becomes far more subtle and intriguing.
Consider the familiar additive groups ℤ, ℚ, ℝ, ℂ, and ℝⁿ, as well as the multiplicative groups of nonzero rationals, reals, complex numbers, and the unit circle group T. Which of these might be isomorphic to a proper subgroup of themselves?
What about your own favourite infinite group from your first modern algebra course? Textbooks often develop standard techniques for showing that two groups are not isomorphic- differing cyclic properties, cardinalities, or the number of elements of a given order etc. Conversely, in introductory courses, demonstrating that two groups are isomorphic typically involves constructing explicit isomorphisms, and many classical examples are transparently isomorphic by design.
The goal of this talk is to illustrate how the question of proper subgroup isomorphisms can enrich an introductory course, and to demonstrate how ideas from Linear algebra can reveal non-transparently isomorphic groups.
