Question

List three Sylow 2-subgroups of ${\mathbf{S}}_{\mathbf{4}}$.

Short answer

Three sylow 2-subgroups of ${\mathrm{S}}_4$are:

$\{\left(1\right),\left(1234\right),\left(12\right)\left(34\right),\left(13\right),\left(24\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1432\right)\}$

$\{\left(1\right),\left(14\right),\left(23\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1234\right),\left(1342\right)\}$ and

$\{\left(1\right),\left(12\right),\left(34\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1423\right),\left(1234\right)\}$.

Step-by-step solution

Step-by-Step SolutionStep 1: Sylow p-Subgroup

Let $G$ be a finite group and p is a prime. If $p^k$ is the largest power of p that divides $\left|G\right|$, then a subgroup of $G$of order $p^k$ is called a Sylow p-subgroup.

Consider a group $S_4$.

Find the three Sylow 2-subgroups of S4

Find the order of $S_4$as:

$\begin{array}{c}4!=24\\ =2^3\cdot 3\end{array}$

Therefore, any subgroup of order role="math" localid="1652855935543" $2^3$is a sylow 2-subgroup of $S_4$.

Now, three sylow 2-subgroups of $S_4$are:

$\{\left(1\right),\left(1234\right),\left(12\right)\left(34\right),\left(13\right),\left(24\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1432\right)\}$

$\{\left(1\right),\left(14\right),\left(23\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1234\right),\left(1342\right)\}$and

$\{\left(1\right),\left(12\right),\left(34\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1423\right),\left(1234\right)\}$

Hence, three sylow 2-subgroups of $S_4$are:

$\{\left(1\right),\left(1234\right),\left(12\right)\left(34\right),\left(13\right),\left(24\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1432\right)\}$,

$\{\left(1\right),\left(14\right),\left(23\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1234\right),\left(1342\right)\}$ and

$\{\left(1\right),\left(12\right),\left(34\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(1423\right),\left(1234\right)\}$.