Finite groups with at least 3 elements cannot have a subgroup consisting of all but one element

Let G be a finite group with |G| = n > 2. Prove that G cannot have a subgroup H such that |H| = n-1.

Under these conditions, there exists a nonidentity element x \in H and an element y \notin H. Consider the product xy. If xy \in H, then since x^{-1} \in H and H is a subgroup, y \in H, a contradiction. If xy \notin H, then we have xy = y. Thus x = 1, a contradiction. Thus no such subgroup exists.

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