Vtgyjfy

I have studied direct products. I know a few applications of direct products, like group isomorphism, etc. What are some applications of sub-direct product of groups?

Subdirect products arise naturally as intransitive subgroups of $S_n$, where they are subdirect products of the induced actions of the group on its orbits. Similarly for completely reducible linear groups.

A powerful tool in group theory is something called the fibre product. This is a special type of subdirect product of the group with itself.

Associated to a short exact sequence $1rightarrow Nrightarrow Hrightarrow Qrightarrow 1$, where $pi(H)=Q$ is the natural map, is the fibre product $Psubset Htimes H$,
$$
P:={(h_1, h_2)mid pi(h_1)=pi(h_2)}.
$$

Clearly the diagonal component $Delta:={(h, h)mid hin H}$ is contained in $P$, so $P$ is a subdirect product.

The first application of fibre products which is know of is that there exist finitely generated subgroups of $F_2times F_2$ which have insoluble membership problem: Firstly, let $Q$ be finitely presented and have insoluble word problem, then $P$ has insoluble membership problem. To see that $P$ is finitely generated, since $Q$ is finitely presented we have that $N subset H$ is finitely generated as a normal subgroup, and then to obtain a finite generating set for $P$ one chooses a finite normal generating set for $Ntimes{1}$ and then appends a generating set for the diagonal $Deltacong H=F_2$.

A surprisingly powerful result, called the $1$-$2$-$3$ theorem, gives conditions on the fibre products to be finitely presentable, see: G. Baumslag, M.R. Bridson, C.F. Miller III, H. Short, Fibre products, non-positive curvature, and decision problems, Comm. Math. Helv. 75 (2000), 457–477.

A super-powerful application of fibre products is the following (the application is the main result of a paper in the Annals of Mathematics, one of the top journals - undisputed top 4 journal, disputed no. 1). If $Gamma$ is a group then $widehat{Gamma}$ denotes its profinite completion.

Question (Grothendieck, 1970). Let $Gamma_1$ and $Gamma_2$ be finitely presented, residually finite groups and let $u :Gamma_1rightarrow Gamma_2$ be a homomorphism such that $widehat{u} :widehat{Gamma}_1rightarrow widehat{Gamma}_2$ is an isomorphism of profinite groups. Does it follow that $u$ is an isomorphism from $Gamma_1$ onto $Gamma_2$?

Theorem (Bridson, Grunewald, 2004, Ann. Math., link). There exists a short exact sequence $1rightarrow NrightarrowGammarightarrow Qrightarrow 1$ where $Gamma$ has a whole host of nice properties, and where $P$ and $Gamma$ are a counter-example to Grothendieck's question: $P$ and $Gamma$ are finitely presentable, with $widehat{P}cong widehat{Gamma}$ but $Pnotcong Gamma$.

Typically, when you know that an object is a subdirect product of other objects, you know that it inherits all properties that are passed from products of those objects to their subsets. For example, a subdirect product of abelian groups must be abelian (and the same for any other group identity).