Why does a filtration of a group consist of normal subgroups, and not any subgroups?

Question

See here:

In algebra, filtrations are ordinarily indexed by $\mathbb {N}$ , the set of natural numbers. A filtration of a group $G$, is then a nested sequence $G_{n}$ of normal subgroups of $G$ (that is, for any $n$ we have $\displaystyle G_{n+1}\subset G_{n}$.

The definition given there does not seem to require the subgroups to have any additional properties:

In mathematics, a filtration $\mathcal {F}$ is an indexed set $S_{i}$ of subobjects of a given algebraic structure $S$, with the index $i$ running over some index set $I$ that is a totally ordered set, subject to the condition that $$ \text{if } i\leq j \text{ in } I, \text{ then } S_{i}\subset S_{j} $$

What am I missing?

Answer 1

Morally, you're going to want to quotient by the objects in the filtration. That's why you need them to be normal. In many categories, notably in any abelian category, you can take the quotient of any object by any subobject, so you only need to say that the objects in a filtration are subojects.

The broad rational is that you want to be able to decompose a group into simple building blocks. Recall that a group is simple if it admits no quotient groups. The big theorem here, which I'll state imprecisely but hopefully clearly, is that every finite group can be written as an extension of simple groups. Building a filtration of a group is a way of writing a group in terms of its simple building blocks. For an example, look at this filtration of $D_{28}$ the symmetry group of a regular $28$-gon.

$$ 0 \to \mathbf{Z}_7 \to \mathbf{Z}_{28} \to D_{28} $$

The subsequent quotients are $\mathbf{Z}_7$, $\mathbf{Z}_4$, and $\mathbf{Z}_2$ You can build $D_{28}$ out of these these groups by taking extensions:

• First take $\mathbf{Z}_7$ and $\mathbf{Z}_4$ and take their cross product $\mathbf{Z}_7 \times \mathbf{Z}_4$. This is $\mathbf{Z}_{28}$.
• Next take that $\mathbf{Z}_{28}$ and the $\mathbf{Z}_2$ and take their semi-direct product $\mathbf{Z}_{28} \rtimes \mathbf{Z}_2$, and this is $D_{28}$.

Answer 2

I have to say that I'm a working group theorist (in geometric group theory) and I don't agree with Wikipedia's definition here. If $G$ is a group and someone says "filtration of $G$", I would expect that to mean a sequence $\{G_n\}$ of subgroups (not necessarily normal) with $G_1\leq G_2 \leq \cdots$ such that $\bigcup_n G_n=G$. What Wikipedia calls a filtration I would call a "descending sequence of normal subgroups". There may be other subfields of group theory where Wikipedia's definition is standard, but at the very least some context ought to be added to Wikipedia's statement

Answer 3

There is also the notion of filtration inherited from the theory of valuations and formal groups. Details can be found in Bourbaki [1] and original ideas can be seen e.g. in [Serre][2]. A filtration of a group $G$ (not necessarily commutative) is a family $(G_{\alpha})_{\alpha\in \mathbb{R}}$ (no normality required at this stage) which is decreasing and left continuous, this means (in an equivalent way) that

• For all $\alpha\in \mathbb{R}$, $\cap_{\beta<\alpha}G_{\beta}=G_{\alpha}$

One then defines $G_{\alpha}^+:=\cup_{\alpha<\beta}G_{\beta}$ which is a subgroup (not necessarily normal) of $G_{\alpha}$.
Now, if the filtration is central, which means that

• For all $\alpha,\beta\in \mathbb{R}$, $\mathbf{(}G_{\alpha},G_{\beta}\mathbf{)}\subset G_{\alpha+\beta}$ (commutator subgroup)

we have

1. $G_{\alpha}^+$ is normal in $G_{\alpha}$
2. $G_{\alpha}/G_{\alpha}^+$ is abelian

Then setting $gr_{\alpha}(G):=G_{\alpha}/G_{\alpha}^+$ (additive notation), one can show that the commutator law (on $G$) passes to quotients so that $$ gr(G)=\oplus_{\alpha\in \mathbb{R}}gr_{\alpha}(G) $$ is a $\mathbb{R}$-graded Lie algebra (one recovers the classical $\mathbb{N}$-graded constructions, in this latter case, factors $gr_{\alpha}(G)$ are $(0)$ for $\alpha\notin \mathbb{N}$). For more on induced (on a subgroup) and quotient filtrations, see (Exercise 2 § 4) at the end of the chapter.

[1] Bourbaki, Lie groups and Lie algebras, ch 1-3 (here chapter 2 § 4)
[2]: https://www-fourier.ujf-grenoble.fr/~panchish/ETE%20LAMA%202018-AP/Serre_Lie%20algebras%20and%20Lie%20groups.pdf