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is Ha a subgroup?

I have this question because i feel like the answer should be know, yet my textbook notation makes it look like it is.

Why I think it should only be a set:

Let G = <a> and H = <a^3>. and |a| = 6,.

Then H = {1,a^3}, Ha = {a,a^4} Ha^2 = {a^2,a^5}.

To me it looks like Ha and Ha^2 can not be subgroups because they are not closed under the operation, unless the operation is multiplying the elements by a^3.. I don't know i'm confused.

But then I think they should be considered subgroups, because it goes on to say that Ha = H iff a is in H. So Ha must be a subgroup if it equals a subgroup... Anyway, anyone have insight here?