Quick questions about the definition for Indecomposable group?

Question

The definition of an indecomposable group is defined as a group $G$ that can not be written as a direct product of its proper normal subgroups. In some places i seen on the internet and books, the last part of the definition states that it cannot be written as a direct product of its proper subgroups. I am just wondering which one should I accept as the correct definition. Also does the definition imply that every element of $G$ can not be written as the direct product of some pair of its proper subgroups/normal subgroups or every elements of $G$ can not be written as the direct product of every pair of its proper subgroups/normal subgroups. As an example:

The classic example is that of $S_3.$

The subgroups of $S_3$ are $\{e,\{e,(12)\},\{e,(13)\},\{e,(23)\},\{e,(123)\},\{e,(123), (132)\}.$ $\{e,\{e,(12)\},\{e,(13)\},\{e,(23)\},\{e,(123)\}$ are not normal subgroups and $\{e,(123), (132)\}$ is a normal subgroup of $S_3.$ But what happens when a group $G$ has more than two normal subgroups and more than two non normal subgroups. We can have the case where if I choose two normal subgroups, than $G$ can be written as a direct product. But if I choose one noraml subgroup and one non normal subgroups, then it fails the definition.

Thank you in advance.

Answer 1

We say $G$ is an internal direct product of $N_1$ and $N_2$ if and only if three things happen:

1. $N_1\triangleleft G$ and $N_2\triangleleft G$; (in particular they are subgroups of $G$); and
2. $G=N_1N_2$; and
3. $N_1\cap N_2=\{e\}$.

You need all three to say $G$ is the (internal) direct product of $M$ and $N$. If all three happens, and only if all three things happen, we write $G=N_1\times N_2$.

A group $G$ is decomposable if and only if there exist nontrivial normal subgroups $N_1$ and $N_2$ of $G$ such that $G=N_1\times N_2$ (internal direct product).

But note that if $G=N_1\times N_2$, then this requires $N_1\triangleleft G$, $N_2\triangleleft G$, $G=N_1N_2$, and $N_1\cap N_2=\{e\}$, so we may conclude that $N_1$ and $N_2$ are normal. Thus, we can just say:

$G$ is decomposable if and only if there exist nontrivial subgroups $N_1$ and $N_2$ of $G$ such that $G=N_1\times N_2$.

Because "$G=N_1\times N_2$" implies $N_1\triangleleft G$ and $N_2\triangleleft G$. So to address the extensive back-and-forth in comments: it doesn't matter if you define "decomposable" specifying the subgroups must be normal or not, in either case you have the subgroups are normal.

Second: a group $G$ is indecomposable if and only if it is not decomposable. Negating the definition gives:

A group $G$ is indecomposable if and only if there do not exist nontrivial subgroups $N_1$ and $N_2$ of $G$ such that $G=N_1\times N_2$.

Or

A group $G$ is indecomposable if and only if there do not exist nontrivial normal subgroups $N_1$ and $N_2$ of $G$ such that $G=N_1\times N_2$.

Or

A group $G$ is indecomposable if and only if for any nontrivial normal subgroups $N_1$ and $N_2$ of $G$, either $G\neq N_1N_2$ or $N_1\cap N_2\neq\{e\}$.

Or

A group $G$ is indecomposable if and only if for any subgroups $N_1$ and $N_2$ of $G$, at least one of the following happens:

1. $N_1=\{e\}$ or $N_2=\{e\}$ (they are not nontrivial); or
2. $N_1$ is not a normal subgroup of $G$, or $N_2$ is not a normal subgroup of $G$; or
3. $N_1\cap N_2\neq\{e\}$; or
4. $N_1N_2\neq G$.

What does this have to do with $G$ having "more than two normal subgroups" or "more than two non-normal subgroups"? It just means you have more pairs to check. If $G$ has only one nontrivial normal subgroup, then either 1 or 3 fails, so there is nothing to do. If it has more than one nontrivial normal subgroup, then you need to check more things. The non-normal subgroups are irrelevant to testing indecomposability, even though the definition need not explicitly mention the two subgrups must be normal, because for $G$ to be decomposable you need two nontrivial normal subgroups with additional properties, regardless of whether you said "normal" or not in the definition.