What is meant by $A\oplus B \oplus C$?

Question

For example $\Bbb Z_2 \oplus \Bbb Z_3 \oplus \Bbb Z_5 $. How should I proceed from . . .

$$\begin{align} \Bbb Z_2 \oplus \Bbb Z_3&= (0,1) \oplus (0,1,2)\\ &=((0,0),(0,1),(0,2),(1,0),(1,1),(1,2))? \end{align}$$

Should I conclude that $$\begin{align} \Bbb Z_2 \oplus \Bbb Z_3 \oplus \Bbb Z_5&= ((0,0),(0,1),(0,2),(1,0),(1,1),(1,2)) \oplus \Bbb Z_5\\ & = ((0,0),(0,1),(0,2),(1,0),(1,1),(1,2)) \oplus (0,1,2,3,4)\\ & = (((0,0),0),((0,0),1)...? \end{align}$$

But that looks terribly inconvenient and messy just for a simple example. What should it be, with examples of how to interpret a sum of more than two groups?

Answer 1

First, use set braces like this: $$\mathbb Z_2 \oplus \mathbb Z_3 = \{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)\} $$ Second, what's inconvenient is to list out every element of the set. That's entirely unnecessary even for the previous example. Instead, write it a bit more compactly like this: $$\mathbb Z_2 \oplus \mathbb Z_3 = \{(i,j) \mid i \in \mathbb Z_2, j \in \mathbb Z_3\} $$ For three direct summands, use ordered triples, just like you would do for points in $\mathbb R^3 = \{(x,y,z) \mid x \in \mathbb R, y \in \mathbb R, z \in \mathbb R\}:$ $$\mathbb Z_2 \oplus \mathbb Z_3 \oplus \mathbb Z_5 = \{(i,j,k) \mid i \in \mathbb Z_2, j \in \mathbb Z_3, k \in \mathbb Z_5\} $$ For a general sequence of groups $G_1,\ldots,G_n$, use ordered $n$-tuples, just like you would for points in $\mathbb R^n$: $$G_1 \oplus \cdots \oplus G_n = \{(g_1,\ldots,g_n) \mid g_i \in G_i \quad\text{for each $i=1,...,n$}\} $$ For all of this, one simply has to keep in mind the general concept of an ordered tuple that one learns in set theory.

Answer 2

One exercise in Dummit and Foote is to show that for sets and groups that $(A\times B)\times C$ is isomorphic to $A\times (B \times C)$. With this in hand you can ignore the parenthesis and consider their elements as tuples of the form $(a,b,c)$. This shouldn't be surprising because as you've noticed, the notation isn't relevant to the group structure in any meaningful way.

Another approach to ordered tuples is to instead consider an indexing set $I$ which define the "coordinates" of the tuple and by considering all functions from $I$ to the sets you're constructing the tuple from. For example if we want to consider $\mathbb{R}^3$ we would take an indexing set containing the letters $\{x,y,z\}$ for the familiar $x,y$ and $z$ coordinates and consider the family $\mathbb{R}^I$ of all functions $f:I \rightarrow \mathbb{R}$. Each of these functions has an $x,y$ and $z$ coordinate given by $f(x),f(y) $ and $f(z)$ respectively. Choose a different function $f' \in \mathbb{R}^I$ it will represent a different point in $\mathbb{R}^3$. This has the advantage of making generalizations to infinite indexing sets very natural as well as allowing you to consider various orders on the indexing set independently from the tuple they represent.

Answer 3

Another approach is via presentations.

If each group $G$ is given by the presentation

$$\langle X_G\mid R_G\rangle,$$

for $X_G$ pairwise disjoint, then one can interpret $A\oplus B\oplus C$ as given by the presentation

$$\langle X_A\cup X_B\cup X_C\mid R_A\cup R_B\cup R_C\cup\{xy=yx\mid x\in X_G, y\in X_H, G\neq H\}\rangle.$$