a) If a group has order 12 , then G contains an element of order 6.
b) If a group has order 18, then G contains an element of order 3.
If it's true, prove it; if it's false find a counter example.
I know I should know this but for some reason can't get my head around it. Can someone give me a push in the right direction. Do I have to include the Sylow Theorems?
The elements are all reflections and they have order 2.
While the order of is .
To have order 6 we require thus .
This means only and have order .
But what is the problem? The above example is when it is true. But the example with is when it is false. Thus, this statement cannot be true i.e. order 12 subgroups has elements of order 6.