Why is the Sylow 2-subgroup of $SL(2,3)$ normal? I know that $n_2 \in \{1,3\} $, where $n_2$ is the number of Sylow 2-subgroups. But how do I show that $n_2 \neq 3$?
[Math] The Sylow 2-subgroup of $SL(2,3)$.
finite-groupssylow-theory
finite-groupssylow-theory
Why is the Sylow 2-subgroup of $SL(2,3)$ normal? I know that $n_2 \in \{1,3\} $, where $n_2$ is the number of Sylow 2-subgroups. But how do I show that $n_2 \neq 3$?
Best Answer
Suppose that there are $3$ Sylow $2$-subgroups, in this case there are at least $3\cdot(8-4)+4=16$ elements of order a power of $2$. Now observe that there are $4$ Sylow $3$-subgroups, so there are $4\cdot(3-1)=8$ elements of order $3$. Therefore every element of $SL(2,3)$ has order $3$ or a power of $2$, ( because $16+8=24$). But the element $$\begin{pmatrix} -1 & -1 \\ 0 & -1 \end{pmatrix}$$ has order 6.