We known that , A nonempty subset H of a group G is a subgroup if
ab−1∈H when ever a and b in H .
In view of the above theorem ,we need only prove that a−1∈H whenever a∈H .
If a=e, then a−1=a and we are done.If a=e consider the sequence. a,a2,a2,........ By closure ,all of these elements belongs to H .Since H is finite ,not all of these elements are distinct.
Say ai=aj and i>j. Then ai−j=e and since a=e,i−j>1. Thus ,aai−j−1=ai−j=e and ,therefore,ai−j−1=a−1. But i−j−1≥1 implies ai−j−1∈H and we are done.
If H is not finite then we have a counter example.
Let G=Z (set of integer) group under addition and H=N (set of natural number).
Then a+b∈N ∀ a,b∈N
But N is not a group under addition.
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