Composition Series of Group of Prime Power Order

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Theorem

Let $G$ be a group whose identity is $e$, and whose order is a prime power:

$\order G = p^n, p \in \mathbb P, n \ge 1$


Then $G$ has a composition series:

$\set e = G_0 \subset G_1 \subset \ldots \subset G_n = G$

such that $\order {G_k} = p^k$, $G_k \lhd G_{k + 1}$ and $G_{k + 1} / G_k$ is cyclic and of order $p$.


Proof

To be proved by induction on $n$.

Let $P_n$ be the proposition for $\order G = p^n$.


Basis for the Induction

$P_1$ is trivially true because:

$\set e = G_0 \subset G_1 = G$

From Prime Group is Cyclic, a group whose order is prime is cyclic.


Induction Hypothesis

Suppose $P_k$ is true for all groups of order $p^k$ for all $k < n$.

We need to show that this implies $P_{k + 1}$ is true.


Induction Step

Let $G$ be a group of order $p^n$.

By Prime Power Group has Non-Trivial Proper Normal Subgroup, $G$ has a proper non-trivial normal subgroup.

There will be a finite number of these, so we are free to pick one of maximal order.

We call this $H$, such that $\order H = p^t, t < n$.

We need to show that $t = n - 1$.


Suppose $t < n - 1$.

Then $G / H$ is a group of order $p^{n - t} \ge p^2$.

Again by Prime Power Group has Non-Trivial Proper Normal Subgroup, $G / H$ has a proper non-trivial normal subgroup, which we will call $N$.

Let $H' = \set {g \in G: g H \in N}$.

We now show that $H \lhd G$.


Let $g, g' \in H'$.

Then $g H, g' H \in N$.

Since $N < G / H$:

$\paren {g H} \paren {g' H} = g g' H \in N$

and so $g g' \in N$.


If $g \in H$, then $g H \in N$.

Since $N < G / H$:

$\paren {g H}^{-1} = g^{-1} H \in N$

and so $g^{-1} \in H'$.

Next:

\(\displaystyle \paren {H'}^a\) \(=\) \(\displaystyle \set {g \in G: a g a^{-1} \in H'}\)
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: a g a^{-1} H \in N}\)
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: \paren {a H} \paren {g H} \paren {a H}^{-1} \in N}\)
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: \paren {g H} \in N^{a H} }\) Definition of Conjugate of Group Subset
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: \paren {g H} \in N}\) as $N$ is normal
\(\displaystyle \) \(=\) \(\displaystyle H'\)


So clearly $H' / H = N$, therefore:

$\dfrac {\order {H'} } {\order H} = \index {H'} N = \order N \ge p$


So:

$\order {H'} \ge p \order H$

contradicting the maximality of $\order H$.

It follows that $t = n - 1$.


Finally, we set $G_{n - 1} = H$ and use induction to show that $P_{n - 1}$ holds.

Since $G / H = G / G_{n - 1}$ is a group of order $p$, then it is automatically cyclic.

$\blacksquare$


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