Group is a union of proper subgroups iff. it is non-cyclic

[gummz]

Homework Statement

Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic.

Homework Equations

None

The Attempt at a Solution

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If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one element of G that is not in that particular subgroup. But then we know that none of the subgroups can represent all the elements of G. Therefore, G is not cyclic.

" <= "
If the group is not cyclic, then no element a in G generates G. That means that G is the union of all the <a> subgroups for all a in G.

Is this correct?

 

[fresh_42]

Looks fine to me.