Question

List the Sylow 2-subgroups and Sylow 3-subgroups of A₄.

Short answer

The Sylow 2-subgroup of A₄ is$\left\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\right\}$ and the Sylow 3-subgroup of A₄ are$\left\{\left(1\right)\left(124\right)\left(142\right)\right\},\left\{\left(1\right)\left(124\right)\left(142\right)\right\},\left\{\left(1\right)\left(134\right)\left(143\right)\right\}and\left\{\left(1\right)\left(234\right),\left(243\right)\right\}.$

Step-by-step solution

Alternative group An

$A_n=\left\{collectionofallevenpermutationsof{S}_n\right\}$

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.

Find Sylow 2-subgroups and Sylow 3-subgroups

Note that, Order of $A_n=\frac{n!}{2}$.

Therefore, order of $A_4=\frac{4!}{2}$.

Simplify the equation as:

$\begin{array}{rcl}{A}_4& =& \frac{4!}{2}\\ & =& 12\\ & =& 2^2.3\end{array}$

Therefore, any subgroup of order 2² is a Sylow 2-subgroup of A₄ and any subgroup of order is a Sylow 3-subgroup of A₄.

Here, there is 1 Sylow 2-subgroup of A₄ and it is written as:

role="math" localid="1659522267584" $\left\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\right\}$

Also, there are 4 Sylow 3-subgroups of A₄ and it is written as:

role="math" localid="1659522432681" $$\begin{gathered} \left\{\left(1\right),\left(123\right),\left(132\right)\right\} \\ \left\{\left(1\right)\left(124\right)\left(142\right)\right\} \\ \left\{\left(1\right)\left(134\right)\left(143\right)\right\} \\ \left\{\left(1\right),\left(234\right),\left(243\right)\right\} \end{gathered}$$

Hence, the Sylow 2-subgroup of A₄ is $\left\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\right\}$ and the Sylow 3-subgroup of A₄ are $\left\{\left(1\right)\left(124\right)\left(142\right)\right\},\left\{\left(1\right)\left(124\right)\left(142\right)\right\},\left\{\left(1\right)\left(134\right)\left(143\right)\right\}and\left\{\left(1\right),\left(234\right),\left(243\right)\right\}$.