Perspectives on Abstract Algebra

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Perspectives on Abstract Algebra
Sebastian Orsted
November 26, 2016
I
n this note, I expand upon the coverage of a selected number of concepts
from Lauritzen (2003). The main purpose is to provide additional examples
and explanations, hopefully providing the reader with more intuition about
the abstract constructions we deal with in the subject. I plan to expand the
notes throughout the course as I notice more aspects of the theory I wish to
emphasize further. The newest version will always be available at
http: //home. math. au. dk/sorsted/persp_algebra. pdf
I shall allow myself to use some notation which is universal standard: I
write φ: X ,→ Y to indicate that the map φ is injective. The hooked arrow is
meant to suggest an arrow put together with a subset symbol “  ”. This makes
sense because an injective map can be thought of as a kind of inclusion; indeed,
notice, for instance, that for an injective group homomorphism i: G ,→ H , the
image Im ( i )  H is a subgroup isomorphic to G via i . Because of this, injective
homomorphisms are often called embeddings . Similarly, we write φ: X ? Y
to indicate that the map φ is surjective. My best guess is that this notation is
meant to suggest “hit Y extra hard”.
1

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