Show <a in D8 : a^2=1> is not a group

[Daniiel]

Hey,

I've been trying to solve this question,

Show that <a in D8 : a²=1> is a not a group.

I might not be processing it properly, but my interpretation of the question is that

<a in D8 : a²=1> = <(a,b): a²=e, b²=2, ab=a^-1b>

Which is just D4, a group, the set of dihedral where 2n=4,

Is the proper interpretation supposed to be

<a in D8 : a²=1> = <(a,b): a²=e, a⁴=e,b²=2, ab=a^-1b>

which is again D4,

Could anyone help me interoperate the question

 

[jgens]

Notice that D₈ = {e,r,r²,r³,s,sr,sr²,sr³}. Now check which of these elements have order 2. Then find two elements of order 2 which when multiplied together do not produce an element of order 2.