Sylow Theorems/Examples

Examples of Use of Sylow Theorems

Sylow $2$-Subgroups in Group of Order $12$

In a group of order $12$, there are either $1$ or $3$ Sylow $2$-subgroups.


Sylow $3$-Subgroups in Group of Order $12$

In a group of order $12$, there are either $1$ or $4$ Sylow $3$-subgroups.


Sylow $7$-Subgroups in Group of Order $28$

In a group of order $28$, there exists exactly $1$ Sylow $7$-subgroup.

This Sylow $7$-subgroup is normal in $G$.


Sylow $2$-Subgroups in Group of Order $32$

The Sylow Theorems tell us nothing about the Sylow $2$-subgroups in a group of order $32$.


Sylow $2$-Subgroups in Group of Order $48$

In a group of order $48$, there are either $1$ or $3$ Sylow $2$-subgroups.

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