In group theory, we say that a group $(G, \cdot)$ is abelian if (and only if) $a \cdot b = b \cdot a$ for all $a, b \in G$.
Abelian groups have certain properties that are generally (that is in the mathematical sense – meaning always) true, so if we know that a group we are dealing with is abelian, we can instantly deduce a variety of things about the group and how it behaves.
Once we know a group is abelian, we can make some useful deductions. Does it work the other way around? Are there truths that will allow us to conclude that a group is abelian?
For what statements are both of these true? What statements imply, and are implied by, the fact that a group is abelian? Essentially, we are asking:
What properties are equivalent to abelianness?
From here onwards we’ll use juxtaposition (like $ab$) rather than an explicit symbol (like $a \cdot b$) to indicate the group operation acting on elements $a$ and $b$.
As an example, take $$(xy)^2 = x^2 y^2 \;\forall x, y \in G$$ Clearly, if G is abelian this is true. Why? Because $$(xy)^2 = (xy)(xy) = x(yx)y = x(xy)y = (xx)(yy) = x^2 y^2$$ using the fact that the group operation is associative and $G$ is abelian.
However, if we write out our first condition slightly differently
$$xyxy = xxyy$$
and then apply the ‘cancellation property’ of groups (cancelling the leftmost $x$ and the rightmost $y$), we end up with
$$yx = xy$$
Succinctly, we have proven
$$(xy)^2 = x^2 y^2 \iff xy = yx$$
for arbitrary elements $x,y$ of a group $G$. Since the elements are arbitrary, we have proven that $G$ is abelian if and only if our original condition is true.
In fact, we might as well define a group to be abelian if it satisfies this property. It may seem strange, but this doesn’t change anything about the mathematics at all. Which definition of the long (and in some sense infinite) list of equivalent definitions we happen to choose to be ‘the definition’ is an entirely human distinction. We can take whichever one we like, and the others are then just consequences of this definition (theorems in our theory).
How about another example? Define the direct product of groups $(G, \circ_{G})$ and $(H, \circ_{H})$, denoted $G \oplus H$, to be the cartesian product $G \times H = \{ (g,h) : g \in G, h \in H \}$ with the group operation $\circ$ defined such that $(g_1, h_1) \circ (g_2, h_2) = (g_1 \circ_{G} g_2, h_1 \circ_{H} h_2)$.
Clearly, the direct product of some number of cyclic groups is abelian, since the cyclic groups themselves are abelian, and the componentwise multiplication of the direct product reduces abelianness in the product to abelianness in each participating group.
It is a much deeper fact that the converse is true too: every (finitely generated) abelian group, say $G$, is isomorphic to the direct product of some number of cyclic groups: this is the so-called Fundamental Theorem:
$$G \cong \bigoplus_{i=1}^{n} \mathbb{Z}_{k_i}$$
where $k_1, k_2, \dots, k_n$ are prime powers.
Again, we might as well define an abelian group to be one that has this form, and then the fact that an abelian group’s elements all commute with each other is just another provable theorem.
I think this subtle and perhaps surprising notion of equivalence is rather interesting. It is conceptually easy to prove that two statements are imply each other: just assume one and prove the other, and then repeat the other way around. But to state that the two statements are in fact perfectly equivalent—there exists no universe in which one of these statements is true and the other false—seems like a much stronger thing. And yet it isn’t.
What other interesting equivalences (or alternative definitions) are there in mathematics?
Well, for example: the definition of a prime number.
Most people, if asked, will probably state that a number is prime if and only if it has no factors other than itself and $1$. This seems reasonable, and indeed it is the original motivating definition. However, we can also say that a number $n$ is prime if and only if whenever $n$ divides $ab$, it divides $a$ or $b$ (or both). The first implication (that this is true if $n$ is prime) is a classical theorem called Euclid’s lemma, but in fact the converse is true too and so the statements are equivalent. We might as well define prime numbers this way.
However, something interesting happens when we move from the integers to something more general. In a general ring, these apparently equivalent definitions separate into two distinct notions called ‘prime’ and ‘irreducible’. It just so happens that in the integers (and in general, in a certain class of rings called GCD domains) they coincide.
Another famous example of an interesting equivalence is of course due to the French mathematican Augustin-Louis Cauchy.
The standard definition of convergence of a real sequence is the following:
A sequence $(a_n): \mathbb{N} \to \mathbb{R}$ is said to converge (to $L \in \mathbb{R}$) if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that for all $n > N$ we have $\lvert a_n – L \rvert < \epsilon$.
In words, it is saying that a sequence converges if and only if we can make the value of the sequence arbitrarily close (as close as we like) to the limit point $L$ by moving sufficiently far along the sequence towards infinity.
There are some functions that satisfy this definition for some (unique!) value of $L$, and some that don’t for any. Another similar but subtly different property a function might have is the following:
A sequence $(a_n): \mathbb{N} \to \mathbb{R}$ is said to be Cauchy if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that for all $n, m > N$ we have $\lvert a_n – a_m \rvert < \epsilon$.
This one is saying, in words, that a sequence is Cauchy if and only if the terms in the sequence can be made arbitrarily close to each other by moving sufficiently far along the sequence.
As you may have suspected, these two properties turn out to imply each other: convergent sequences are Cauchy, and Cauchy sequences converge. The first implication is the easier one to prove, but both are true and therefore the two properties are equivalent. They are both perfectly good characterisations of convergence. This is particularly convenient since they contain an important conceptual difference: the second definition makes no mention of a limit, so we can prove convergence to some limit even when we don’t know what that limit actually is. Having these kind of equivalences is actually quite powerful!
So, what is the meaning of all of this? The question is perhaps not a mathematical one, and even then I’m not exactly sure of the answer.
My impression is this: definitions are less important than often assumed; it is the underlying properties of mathematical objects that are fundamental. Definitions are just a human way of putting an abstract concept into one-to-one correspondence with a more concrete ‘testable’ truth.
There are some sequences that are Cauchy, and some sequences that converge. In fact, we are talking about the same set of sequences that all have some ‘underlying property’ in common, but as humans we like to anchor this property to things we can more readily pin down in symbols—even if we end up doing so from two quite different angles.