The maximal subgroups of the symmetric group $S_n$ are fundamental to understanding its structure and how its elements can be rearranged while preserving certain properties. These subgroups represent the largest possible proper subgroups within $S_n$, meaning they are not contained within any other subgroup apart from $S_n$ itself.
Understanding Maximal Subgroups
A subgroup $H$ of a group $G$ is considered a maximal subgroup if $H$ is not equal to $G$, and there is no other subgroup $K$ such that $H \subsetneq K \subsetneq G$. In simpler terms, you cannot expand a maximal subgroup $H$ by adding any element from $G$ without forming the entire group $G$.
Classification of Maximal Subgroups of $S_n$
For $n \ge 3$, the maximal subgroups of the symmetric group $S_n$ are primarily categorized into three main types, derived from the O'Nan–Scott Theorem which classifies primitive permutation groups. These types depend on how the subgroups act on the set of $n$ elements being permuted.
1. Intransitive Subgroups
These subgroups do not act transitively on the set of $n$ elements; they preserve a non-trivial partition of the set into two disjoint, non-empty subsets. This means the elements within each subset are permuted independently.
- Structure: They are isomorphic to $Sk \times S{n-k}$ for $1 \le k < n/2$. They permute elements within a chosen subset of size $k$ and independently permute elements within the remaining $n-k$ elements.
- Example: The most common form is the point stabilizer, which fixes one specific element and permutes the other $n-1$ elements. These are isomorphic to $S_{n-1}$. For $S_n$, there are $n$ such subgroups (one for each element fixed), and their index in $S_n$ is $n$.
- For $S_4$, the point stabilizers are isomorphic to $S_3$. There are 4 such subgroups, each fixing one of the 4 elements (e.g., the subgroup fixing '1').
2. Imprimitive Subgroups
These subgroups act transitively on the set of $n$ elements but preserve a non-trivial partition of the set into $k$ blocks of equal size $m$, where $n=km$ and $k,m > 1$. The subgroups permute elements within each block and also permute the blocks themselves.
- Structure: They are isomorphic to the wreath product $S_m \wr S_k$.
- Action: They organize the $n$ elements into $k$ distinct blocks, each containing $m$ elements. Permutations can happen within each block, and the blocks themselves can be rearranged.
- Example: For $S_6$, a subgroup preserving a partition into 3 blocks of 2 elements (e.g., ${{1,2}, {3,4}, {5,6}}$) would be isomorphic to $S_2 \wr S_3$.
- Special Case of $S_4$: For $S_4$ ($n=4$), we can partition the 4 elements into 2 blocks of 2 elements ($k=2, m=2$). The maximal subgroups of this type are isomorphic to $S_2 \wr S_2$. These are precisely the three Sylow 2-subgroups of $S_4$, each of order 8 (e.g., the dihedral group $D_4$).
3. Primitive Subgroups
These are subgroups that act transitively on the set of $n$ elements and do not preserve any non-trivial partition into blocks. This category includes two sub-types:
- The Alternating Group ($A_n$): For any $n \ge 2$, the alternating group $A_n$ (the subgroup of all even permutations) is always a maximal subgroup of $S_n$. Its index in $S_n$ is always 2.
- For $S_4$, $A_4$ is a maximal subgroup.
- Other Primitive Groups: For $n \ge 5$, there are additional maximal subgroups that are primitive but do not contain $A_n$. These groups are often more complex, arising from specific constructions like affine groups $AGL(d, p)$ when $n=p^d$ (acting on the points of an affine space), or from nearly simple groups. These "sporadic" types are less common for larger $n$ but complete the classification.
The Case of $S_4$: A Detailed Example
The symmetric group $S_4$ serves as an excellent illustration of these classifications, as it encompasses all the primary types of maximal subgroups:
- Point Stabilizers (Intransitive): These are the 4 subgroups isomorphic to $S_3$. Each fixes one element and permutes the other three.
- Alternating Group $A_4$ (Primitive): This is the subgroup of all 12 even permutations in $S_4$.
- Sylow 2-subgroups (Imprimitive): These are the 3 subgroups isomorphic to the dihedral group $D_4$ (order 8). Each preserves a partition of the 4 elements into two blocks of two (e.g., ${{1,2}, {3,4}}$).
Index of Maximal Subgroups
A notable property concerning maximal subgroups of $S_n$ for $n \ge 3$ is related to their index. If $H$ is a maximal subgroup of $S_n$ and $H$ is not the alternating group $A_n$, then its index $|S_n : H|$ is generally greater than or equal to $n$.
However, there is a specific exception to this rule for $n=4$:
- The Sylow 2-subgroups of $S_4$ are maximal, but their index in $S_4$ is 3 (since $|S_4|=24$ and $|D_4|=8$), which is less than $n=4$. This exception underscores the unique group-theoretic properties of $S_4$.
Summary Table of Maximal Subgroup Types
| Type of Maximal Subgroup | Description | General Example for $S_n$ ($k, m > 1$) | Example for $S_4$ |
|---|---|---|---|
| Intransitive | Preserves a partition into two disjoint sets. | $Sk \times S{n-k}$ (e.g., $S_{n-1}$) | Point stabilizers ($S_3$) |
| Imprimitive | Preserves a partition into $k$ blocks of $m$ elements ($n=km$). | $S_m \wr S_k$ | Sylow 2-subgroups ($D_4 \cong S_2 \wr S_2$) |
| Primitive ($A_n$) | The alternating group, containing all even permutations. | $A_n$ | $A_4$ |
| Primitive (Other) | Other primitive groups not containing $A_n$. | $AGL(d,p)$ when $n=p^d$, nearly simple groups. | (None distinct from above) |
This classification provides a comprehensive framework for understanding the internal structure of symmetric groups, which are fundamental objects in abstract algebra.
