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Previously, we gave a rapid overview of short exact sequences. In this post, I’ll go into more detail about what the things appearing in a short exact sequence actually are.

This post is incomplete! I have a bit more to add.

An abelian group is a setting (or “data type”) where we can add and subtract.

Numerical examples

The starting examples are

  • the real numbers $\R$, e.g. $0, -8.91, 1.5, \sqrt2, \dotsc$
  • the integers $\Z$, i.e. $\dotsc,-2,-1,0,1,2,3\dotsc$

Here are some non-examples of abelian groups:

  • the natural numbers $\N$, i.e. $0,1,2,\dotsc$ (no negatives). This isn’t a group because you can’t subtract: $2-3$ will “break you out” of $\N$.

  • the set of irrational numbers: $\sqrt 2$ and $1-\sqrt 2$ are both irrational, but their sum $\sqrt2 + (1 - \sqrt 2) = 1$ is rational, so addition will “break you out” of irrational numbers.

Physical examples

Here’s a different kind of example.

  • Physical displacements in one dimension. We can add and subtract physical displacements like so:

    \begin{align*} {\xra{\hspace{2cm}}} + {\xra{\hspace{4cm}}} &= {\xra{\hspace{6cm}}},\\ {\xra{\hspace{2cm}}} - {\xra{\hspace{4cm}}} &= {\xleftarrow{\hspace{2cm}}} \end{align*}

    In a certain sense, this example is just the earlier example $\R$ in disguise. For example, if we measure distances in centimeters, then we can write the above equations as

    \begin{align*} 2 + 4 &= 6,\\ 2 - 4 &= -2. \end{align*}

    (They’re coded as 2/4/etc. cm, but will probably appear bigger on your screen.)

    But we could also measure distances in inches, in which case we’d instead write

    \begin{align*} {0.78...} + {1.57...} &= {2.36...}\\ {0.78...} - {1.57...} &= {-0.79...} \end{align*}

    One reason for the abstract language of abelian groups, rather than just talking about normal addition of numbers, is so that we can describe the intrinsic properties of these examples without making an arbitrary choice of units. It provides “typechecking” that prevents you, for example, from accidentally converting square inches to square centimeters by multiplying by $2.54$ rather than $2.54^2$ (but I won’t explain that bit in this post).

  • Time durations, such as minutes, hours, days, years. For subtraction, you can interpret 1 hour as “1 hour after”, and $-1$ hours as “1 hour before”. Again, this is really $\R$ in disguise if we fix a unit of time, but there’s no best unit to choose.

“Numerical” examples like $\R$ are “models” of the “in-nature” examples above. This helps explain why we care about $\Z$ on its own rather than only talking about $\R$: we could use the equation \[ -3 + 2 = -1 \] to say “I loaned him three ducks, he gave me two back, so he still owes me one duck”. But an equation like \[ 0.5 + 0.5 = 1 \] (he gave me half a duck and another half of a duck, so now I have a whole duck) is nonsensical, assuming the ducks are to be kept alive. That is, many situations are modelled by $\Z$ rather than $\R$.

From the modern point of view, the $1$ living in $\Z$ is a different mathematical object than the $1$ living in $\R$ (the latter can interact with numbers like $0.5$, the former can’t). Of course, they “behave exactly the same way”—we encode this using maps.

Wrap-around examples

The next class of examples are where things start to get more interesting. In many situations, we want or need to “throw away” information.

  • The cyclic group of order $n$ consists of the numbers $0, 1, …, n-1$ with “wrap-around addition”: we treat $n$ as $0$. It’s denoted by $\Z/n\Z$, or just $\Z/n$ if I’m being lazy (which I usually will be), or also $C_n$.

    • in $\Z/4\Z$ we can write \[ 3 + 1 = 4 = 0, \] or \[ 3 = -1. \] (Think of $-1$ as “the thing which when added to $1$ gives $0$”, rather than “$1$ with the opposite sign”.) We could use this equation to describe “three right turns equals one left turn”.

    • in $\Z/60\Z$, we can write \[ 30 + 45 = 15 \] to talk about minutes elapsed on a clock (ignoring the hours)

    • in $\Z/2\Z$, we can write \[ 1 + 1 + 1 + 1 + 1 = 5 = 3 = 1 \] to indicate that flipping a switch three or five times has the same effect as flipping it just once. You can think of $\Z/2\Z=\{0,1\}$ as consisting of $\{\text{even}, \text{odd}\}.$

  • Angles or rotations form a group denoted by $\T$. If we choose an angular unit and a direction, say degrees and counterclockwise, then we can describe angles using numbers, like \[ 90^\circ \text{ or } {-123^\circ} \] ($-123^\circ$ counterclockwise = $123^\circ$ clockwise). Note that rotating something by $30^\circ$ and then by $40^\circ$ is the same as rotating it by $30+40=70$ degrees, so we can also view $30^\circ$ as “the act of rotating by $30^\circ$” and “addition” in $\T$ as composition (doing one operation and then another).

    But there’s a catch: $-360^\circ,0^\circ,360^\circ,720^\circ,$ etc. all correspond to the same angle, as do $10^\circ,370^\circ,730^\circ,\dotsc$ We could also write $\T$ as “$\R/360\Z$”.