Proving the Inclusion of <a> in H: A Permutation Group Proof

[iamalexalright]

Homework Statement

Suppose G is a group, H < G (H is a subgroup of G), and a is in G.

Prove that a is in H iff <a> is a subset of H.

Homework Equations

<a> is the set generated by a (a,aa,aa^-1,etc)

The Attempt at a Solution

For some reason this seems too easy:

1. Suppose a is in H.
Since H is a group, a^-1 is in H.
Since H is a group aa, is in H (as is aa^-1, etc.)
Thus <a> is a subset of H.

2. Suppose <a> is a subset of H.
Obviously a is in H.

And this completes the proof... or am I missing something?

 

[iamalexalright]

shameful bump