I am trying to understand the definition of subdirect subgroup and full diagonal subgroup. But I am finding it hard to undertstand. Could anyone explain it with example? Particularly when we are to find the subdirect subgroup of a direct product of non abelian simple groups? For example $G=A_5 \times A_5 \times A_5$. What will be it's subdirect subgroup and diagonal subgroup? There is a lemma 1.4.1 in the thesis http://www.joannafawcett.com/masters.pdf on page 12.
I am particularly interested to understand the 2nd part of this lemma which states that Let $G = T_1 \times T_2 \times \cdots \times T_k$ be a direct product of nonabelian, simple groups $T_1,\ldots,T_k$ ($k ≥ 1$). Let $H$ be a subgroup of $G$ and $I := \{1,\ldots, k\}$. If $H$ is a subdirect subgroup of $G$, then $H$ is a direct product $\prod H_j$ , where each $H_j$ is a full diagonal subgroup of some subproduct $\prod_{i \in I_j} T_i$ and $I$ is partitioned by the $I_j$ .
I am looking for it's example. Because I couldn't make sense what does partition means here in the form of $T_i$. For example if we have $G=A_5 \times A_5 \ldots A_5$. Then according to this lemma how H, T_i and I_j look like? How do we construct it by using this lemma?
The diagonal subgroup of $A_5\times A_5\times A_5$ is given by $$ D=\{(g,g,g)\mid g\in A_5\}, $$ and for any epimorphism $\phi_i\colon A_5\rightarrow A_5$, the subgroup of $A_5\times A_5$ given by $\{(g,\phi(g))\mid g\in G\}$ is a subdirect product of $A_5$ and $A_5$. Similarly for $A_5\times A_5\times A_5$. The definition here is good to understand, I think.