?

Free Version

Upgrade subject to access all content

Moderate

Composition Series of a Group: Definition and Basic Properties

ABSALG-GG4XV6

Let $G$ be a group with neutral (identity) element $e$. A subnormal series, and in particular, a composition series, is a way to study $G$ in terms of the normal subgroups of its normal subgroups and the quotients of its normal subgroups by their normal subgroups.

Suppose we have a chain of strict inclusions of subgroups:

$$\{e\}=G_0\vartriangleleft G_1\vartriangleleft G_2\vartriangleleft\ldots\vartriangleleft G_k=G$$

...with $k\ge0$ finite and with factor (quotient) group $G_{i+1}/G_i$ a simple group, $i=0,\ldots, k-1$, for $k\ge 1$.

Here, $G_i\vartriangleleft G_{i+1}$ means that $G_i$ is a normal subgroup of $G_{i+1}$, $i=0,\ldots,k-1$.

Then the chain of subgroups is called a Composition Series. The finite integer $k$ is called the length of the series and the $G_{i+1}/G_i$ are called the composition factors of the series.

Which of the following is FALSE?

Select ALL that apply.

A

Every non-trivial finite group has a composition series.

B

Every infinite group has a composition series.

C

The factors of the composition series of a non-trivial group are isomorphic, up to reordering.

D

Each $G_i$ in a composition series is a maximal normal subgroup of $G_{i+1}$, $i=0,\ldots,k-1$.

E

The subgroups $G_i$ of a composition series are all normal in $G$.