Schreier-Zassenhaus Theorem

Theorem

Let $G$ be a finite group.

Let $\mathcal H_1$ and $\mathcal H_2$ be two normal series for $G$.

Then $\mathcal H_1$ and $\mathcal H_2$ have refinements of equal length whose factors are isomorphic.

Proof

Suppose that:

$(1): \quad \left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_{n-1} \triangleleft G_n = G$

and:

$(2): \quad \left\{{e}\right\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_{m-1} \triangleleft H_m = G$

are two normal series for $G$.

Let a new series be formed:

$(3): \quad \left\{{e}\right\} = \hat G_0 \subseteq \hat G_1 \subseteq \cdots \subseteq \hat G_{n-1} \subseteq \hat G_{n m} = G$

such that:

$\hat G_k = G_q \left({G_{q+1} \cap H_r}\right)$

for $k = q m + r$, where $0 \le q < n$ and $0 \le r \le m$.

Note that, by the above construction:

$\hat G_{q m} = G_q \left({G_{q+1} \cap H_0}\right) = G_{q-1} \left({G_q \cap H_m}\right) = G_q$

We have that $\hat G_k$ is well-defined, and that each group in $(1)$ also appears in $(3)$.

Also, each $\hat G_k$ is a normal subgroup of $\hat G_{k+1}$

$(A): \quad$ But note that $(3)$ may not actually be a normal series as it is possible that $\hat G_k = \hat G_{k+1}$ for some $k$.

Having created series $(3)$, we use the same procedure to form:

$(4): \quad \left\{{e}\right\} = \hat H_0 \subseteq \hat H_1 \subseteq \cdots \subseteq \hat H_{n-1} \subseteq \hat H_{n m} = H$

such that:

$\hat H_k = H_q \left({H_{q+1} \cap G_r}\right)$

for $k = q n + r$, where $0 \le q < m$ and $0 \le r \le n$.

The same statements that were made about $(3)$ also hold about $(4)$.

Now let $k = u m + v$ and $l = v n + u$.

It follows that:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {\hat G_{k+1} } {\hat G_k}\)	\(=\)	\(\displaystyle \frac {G_u \left({G_{u+1} \cap H_{v+1} }\right)} {G_u \left({G_{u+1} \cap H_v}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from above
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\cong\)	\(\displaystyle \frac {H_v \left({H_{v+1} \cap G_{u+1} }\right)} {H_v \left({H_{v+1} \cap G_u}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Zassenhaus Lemma
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\hat H_{l+1} } {\hat H_l}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from above

So $(3)$ and $(4)$ have isomorphic factors.

In $(A)$ it was remarked that some of the factors in these isomorphic series may have redundant elements where $\hat G_k = \hat G_{k+1}$ and similarly for $\hat H_l = \hat H_{l+1}$.

As the series consist of isomorphic elements, all we now have to do is remove these redundant elements from both (they will occur at the same place).

Hence we end up with two refinements of equal length whose factors are isomorphic.

$\blacksquare$

Source of Name

This entry was named for Otto Schreier and Hans Julius Zassenhaus‎.

It is also known as the Schreier Refinement Theorem.

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 72$