Composition Series (unfinished)

Definition A subgroup series for a group $G$ is a chain of normal subgroups that gets down to the trivial group $$1 = G_0 \le G_1\le G_2 \le \ldots \le G_n = G.$$

Definition A series is normal if each $G_i \unlhd G_{i+1}$.

Definition The factors of a normal series are the quotients $G_{i+1}/G_{i}$.

Definition A series if proper if the containments are all strict $G_i \lhd G_{i+1}$.

Definition A series for $G$ is a refinement of another series for $G$, if it's just had zero or more new groups added in, a refinement of a normal series must be again normal. A refinement is proper if it's has at least one new group added.

Definition A composition series is a proper normal series which has no proper refinements. (so $1 \lhd G$ is a composition series for any nontrivial $G$).

Lemma Any proper normal series can be refined to a composition series.
proof: Just keep refining it until you have a composition series, a composition series must be finite since our groups are finite,.

Proposition A normal series $$1 = G_0 \unlhd G_1\unlhd G_2 \unlhd \ldots \unlhd G_n = G$$ is a composition series iff each factor is simple.
proof: ($\Leftarrow$) No factor is trivial. If a factor wasn't simple, i.e. $G_i/G_{i+1}$ has a proper nontrivial normal subgroup $B$ then $B \simeq K/G_i$ for some $G_i \lhd K \lhd G_{i+1}$ (by Noether4) and this gives a proper refinement. ($\Rightarrow$) not done yet.

Definition Two series for $G$ are equivalent if the factors of one are permutation of the factors of another.

Theorem Factors of a composition series are simple.
proof:  Let us have a normal series with $\cdots \unlhd H_i \unlhd H_{i+1} \unlhd \cdots$ where $H_{i+1}/H_i$ isn't simple, say it has some normal subgroup $H_i/H_i \lhd N \lhd H_{i+1}/H_i$, then by (Noether4) lattice isomorphism we get the existence of a subgroup $H_i \lhd N' \lhd H_{i+1}$. A refinement of our series.