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I am self-studying Hungerford's book Algebra. In the page $59$ he wrote the following:

In this section we study products in the category of groups and coproducts in the category of abelian groups. These products and coproducts are important not only as a means of constructing new groups from old, but also for describing the structure of certain groups in terms of particular subgroups (whose structure, for instance, may already be known).

I would like to know why is it important to construct new groups from the old ones?

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How many examples of groups do you know of? – Mariano Suárez-Alvarez Feb 28 '12 at 0:26
@MarianoSuárez-Alvarez: $S_{n}$, $Z_{n}$, $GL(n;R)$, orthogonal groups, Lorentz groups, symplectic groups, Poincaré groups, Heisenberg groups, Euclidean groups, if $\mathbb{F}$ is a field then $\mathbb{F}^{\star}$ is a group. There are a lot of them. – spohreis Feb 28 '12 at 0:33
Presumably you do not think the ones you know are all of them... having various constructions of new groups could possibly be useful in finding new examples, no? (Of course, there are many other ways to construct groups from old ones, which can be used to construct more interesting examples than products and coproducts!) – Mariano Suárez-Alvarez Feb 28 '12 at 0:37
One reason is that all finite abelian groups can be written as the direct sum (coproduct) of cyclic groups of prime order. – Nick Alger Feb 28 '12 at 0:47
I think that it’s a bit misleading of Tom to separate the construction of new groups from old and the description of groups in terms of subgroups: they’re inextricably intertwined as part of the study of the structure of groups. – Brian M. Scott Feb 28 '12 at 0:47
up vote 7 down vote accepted

In applications, you might run into some group that you care about (e.g. it might act as automorphisms on some object that you care about). How do you describe and study this group? Well, probably a good idea is to describe in terms of other groups you understand and use your understanding of those groups to understand this new group.

For example, sometimes you'll run across a finitely-generated abelian group (e.g. because of Dirichlet's unit theorem or the Mordell-Weil theorem). That's great because the structure theorem tells you that all such groups are finite products of cyclic groups, so you can understand them by understanding cyclic groups.

As another example, the modular group $\text{PSL}_2(\mathbb{Z})$ is a group of great importance to the theory of modular forms and related subjects. Surprisingly, it can be described as the coproduct $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$.

A more general class of important examples is computing fundamental groups using the Seifert-van Kampen theorem.

You'll run into more examples like this the more you keep studying mathematics, so my advice is not to seek a definitive list of examples or anything like that. Just trust in the general principle; it is very fruitful.

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One more, cliched but fundamental, example of why it's important to know how groups can be constructed from other groups.

As you know, finite simple groups have been classified into 16 infinite families and an additional 26 groups that don't fit into these families. ( )

A finite group theorist I know describes much of what he does as "tabular mathematics:"

Step 1) Prove that some statement holds for all the finite simple groups. (Here's where he has to go through the table of all such groups, hence "tabular."

Step 2) Prove that if $G\cong H/K$ and the statement holds for $H$ and $K$, then it holds for $G$.

Needless to say, this is a very fruitful approach, and it is more or less the main approach to any program for proving the inverse Galois problem.

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